I've never seen much of an attempt to flesh out a surviving Napoleonic Europe TL anywhere on this site nor for that matter anywhere in literature!
I have to wonder why that is; I'd guess there is a consensus that like Hitler's Europe or Alexander's empire the thing is just too unstable; that it could only be acquired via a mad rush of military exertion that presumably leaves a society too geared to war, too undermined by its wartime expedients, having been thrown together too fast to mind creating properly stable local coalitions for it to be held together other than by main strength--a strength too sure to be dissipated in unending military ventures and challenged by strong powers outside it can never conquer and subdue, powers keen to exploit the many centrifugal forces desiring to tear it apart.
I can see the strength of such arguments but it still strikes me as a remarkable failure of imagination that no one ventures to spell out how and why the majority of Napoleon's conquests might in fact be stabilized as a going concern. The arguments against are many, I suppose, but consider the arguments for:
1) That having settled borders very far away from most populations and harmonizing all deep differences between component states should lead to a Pax Napoleonica; war being eliminated between the numerous former states of Europe, military force can be recruited and supplied on an Empire-wide basis and concentrated along relatively short borders (vast compared to any prior state but short divided by the area within them I mean). Not only is bloodshed greatly reduced, but the economic cost of provisioning an army of overwhelming power along those borders is much reduced versus the Old Regime era cost of provisioning an order of magnitude more soldiers to defend fractal borders two orders of magnitude longer! Or, since the Old Regime 18th century warfare tended to be in a phase where relatively small professional armies were moved like chessmen in a set-piece opera of conventionalized warfare, and thus by a sort of gentleman's agreement the burden of sufficiently large armies was mitigated, an army of truly Napoleonic scale, a far greater share of the population, can be maintained at costs per subject comparable to or a bit lower than the Old Regimes' levies for the purpose.
By related logic, a consolidated Napoleonic system should have the resources to devote to developing a Navy of skill and size sufficient to at any rate punch through the RN blockade and open up world oceans to trade, even if the British persist in unrelenting hostility. More likely, at some point the British throw in the towel, recognize that there exists a continental hegemon they can't dislodge try as they might, and get down to dealing with one another on a peacetime basis--either allowing Empire shipping to develop to compete with British, or undertaking to trade in Imperial ports so the Empire can at any rate purchase globally produced goods, if at something of a disadvantage in terms of trade.
2) That while most of the old regime monarchs and nobility had deep grievances against the Napoleonic system, and there are doubtless many regions where one can point to hostility throughout the populace, other regions were places where Napoleon's forces and schemes were welcomed with some gratitude. The best example that comes to my mind is Poland, but i imagine there must have been others and that sufficiently astute post-conquest era policy could cultivate still more. With enough regions that have a stake in the Napoleonic system providing people on all levels--soldiers, agricultural, mining and industrial workers, and leaders--who support it, the centrifugal tendencies of others can be checked.
3) Although Hitler certainly liked to compare himself to Napoleon, I think we can assume if the downtime conqueror could familiarize himself with the Corporal's biography and nature, he'd angrily reject the comparison, despite his autocratic tendencies. As you note, the Napoleonic system spread a whole slew of Enlightenment projects more or less perfected to workable form by the French revolutionaries and their more conservative successors, mainly Napoleon himself. He is know the continent over as the liberator of Jews for instance, an example that Herr Hitler seems to have overlooked.
On the whole as a modernist I certainly like a continued Napoleonic system better than the Ancient Regime restorations that were attempted afterward in reaction--though it is an open question whether 50 or 60 years on Europe would be better or worse off; the age of reaction did not last all that long before being drowned again in yet other waves of revolution and compromises to head off still others. Might a successful Napoleonic regime early in the century lead to a more effective dead hand of reaction in later generations? Or would progress be facilitated? One would have to game out a detailed ATL to be sure. But there is no doubt in the immediate timeframe of the first couple decades, that the Code Napoleon was preferable in many ways over the old ways or what the disgruntled reactionaries would wish to restore. Politically then it should be the basis of much consensus-building, and economically the many reductions in barriers to trade, assisted by positive visionary development schemes and with the likely negative consequences, foreseen and otherwise, mitigated by Imperial intervention and ad hoc rulings and settlements, should lead to snowballing development, a rising standard of living and thus still more consensus for the Empire.
So your very specific question is hard to answer because while I think as sketched out above that a successful and lasting Napoleonic empire is conceivable enough, still for whatever reason no one has done the homework for us of imagining it in detail, so our speculations as to what the imperial subjects get up to in bed and then where their more or less numerous children wind up going to work and live out their lives is pretty much a wild guess. It sure would be nice if we had some other industrializing European country that also adopted Napoleonic countryside reforms comparable to those that persisted in France (despite the obvious hostility of restored Bourbons and the unclear interests of the Orleanists) and we could thus better judge if French 19th century demographic trends really did stem mainly from their reformed interests as small landholders afraid to have too many children and thus dilute the holdings down below sustainable levels, versus perhaps peculiar French responses and issues. It does seem likely that if the reforms of the French countryside, which bought Napoleon (and whether they appreciated it or not, other French monarchies) considerable stability and loyalty) were replicated systematically all over Europe (the parts Napoleon controlled anyway, which was most of it) that the countryside population growth would slow somewhat, since those landholding considerations are powerful ones. But we can hardly be sure, and how sure are we that the system would in fact be instituted universally? The Empire was in fact not monolithic but a system of large kingdoms and duchies entrusted to Napoleonic supporters under his direction; different kingdoms might adopt different policies according to local circumstances. Such inconsistency is against the unifying and simplifying spirt of the French expansion to be sure, but also a pragmatic privilege of monarchs who don't have to claim legitimacy from an alleged universal will as shown by the polls.
We'd really have to sketch out in some detail exactly how the persistent system survives, whether and to what extent and where it prospers, how stable or unstable it is, and how it is situated in the world as a whole. I haven't looked into your "Quasi-War" installments yet for instance, but I gather the point is that the USA, under stronger Federalist hegemony, turns more sharply than I'd have guessed likely against the French, despite democratic sentiments (muted under Napoleon's dictatorship to be sure) favoring the revolutionary people who also aided our own revolution. If this hostility persists long enough without some U-turn, and the USA remains traditionally hostile to France and hence its Empire, I suppose the upshot is that the world outside of Europe is more dominated even than OTL by the UK, while OTL seems like quite enough of a Brit-Wank as it is! Then again, if the British can't trade with Europe they are worse off in ways that even a complete domination of global trade might not fully compensate.
Realistically, I'd think that the USA would find the Napoleonic empire a natural trade partner and a good counterbalance to British power, which much more immediately threatened the US. Perhaps in a period where Napoleon and his successors keep bidding to reestablish themselves on a global stage this would seem less safe to be sure, but the French will surely still be the underdog away from Europe and American policy ought to therefore favor them as being more willing to make favorable deals for access.
I suppose a possible stable settlement of the European conflict is that the British, still supreme on the seas but uneasily aware that the Empire's resources could construct a war fleet that could overwhelm even the RN, and a merchant fleet to displace the British from their livelihood as merchants to the world, cut their deals with France so as to allow a token face-saving share of both trade and seapower to the continental giant, while reserving the lion's share to themselves, in return for allowing that token to rove the seas in peace, for British trade to dominate but on a fair basis, and perhaps even return some of the colonial holdings France and other conquered powers (the Dutch for instance) held before 1800. Under those circumstances I suspect Yankee policy would favor France to the limit the treaty permitted (still leaving plenty of trade for the British to carry) and welcome Imperial investment in US development (as neither they nor the British could extort more return than contract allowed, nor guarantee their failed investments by force on US soil).
As a first guess, I daresay that emigration from the European continent under the Empire would indeed slow down and be at lower levels, because the Empire would not only produce fewer offspring but also find roles for a greater share of those who are born there. The relative attraction of America would indeed diminish and also, Imperial policy might favor sending more of those who do emigrate to either colonies or chosen client states that might veer to an Imperial alliance in defiance of British power in Latin America.
Britain on the other hand might well be a more repressive, less democratic realm than it developed to be OTL, with the police state mobilized OTL against Nappy persisting and with democratic movements all the more tarred with the brush of foreign subversion. The British "surplus population" might be both larger and more desperate and all the more keen to escape to the USA, or accept a place in the colonies such as Canada--or be transported to Australia.
Certainly, unless we envision the Napoleonic system breaking down (as it might well, unfortunately perhaps) in the later 19th century, there won't be the wave of political immigration fleeing the reactionary backlash to the revolutions of 1848. In particular Germans in that wave were influential in helping found the Republican Party.
To be sure the whole course of the USA, including the slavery question, will probably be much butterflied in a Napoleonic unified Europe TL.
I hope my response will at least provoke someone with stronger views on just what would happen to European demographics and politics to join in, if only to naysay the both of us! Or that you might perhaps (if there aren't lots of Napoleonic victory TLs that I am just ignorant of) pioneer on this site a serious look at what such an ATL might look like.
Honestly, in all of fiction the only ATL with a persisting Napoleonic empire lasting long after Bonaparte's own death I have ever read is found only in Keith Laumer's
Worlds of the Imperium, written in the early '60s. Laumer assumed, on I don't know what basis, that in such a world (achieved by Napoleon invading and conquering Britain, which I think is pretty unlikely) technical progress would slow down to a crawl, so that Europe of the 1960s would have a technical level comparable to OTL 1870s.
I see no reason for such extreme pessimism as that!
And so I encourage you to go ahead and show us Bonapartist Europe as you think it might really be, even if we do already disagree on the question of the US relationship with it.