I voted yes.
Note that since the question is not "would the Empire keep pace or pull ahead?" but "could it compete?" the choice of "maybe" is rather redundant; it means the same thing as "yes" but with a nuance of more hesitation. "Compete" might not even mean doing as well per capita, it might mean doing a lot worse per capita--but the overall region of the Empire (which I for one interpreted to mean the whole Napoleonic-dominated system) is better off than the same region overall OTL.
Seen so broadly, it seems quite likely that the region that for an implied long term, of a generation or more, stays under Napoleonic control would hardly do worse than OTL, unless one believes there is something pathological about the Napoleonic system. OTL the heavy burden of ongoing war doubtless made it worse than it might have been, and there are a number of clearly positive elements Napoleon brought with him.
There is the simple fact of unity itself for instance; that tends to favor development.
The Code Napoleon was developed with the intention of being a rational system of law for the times, and remained influential in France and indeed throughout western Europe OTL despite the eventual collapse of the Empire that promoted it; presumably if that Empire does not collapse and perhaps gradually expands farther or anyway consolidates itself on the 1807 boundaries, not only would this legal system stay in place but it might be amended, upgraded by the same administrative savants who were commissioned to develop it in the first place.
In general, the conquering French tended to "liberalize" any place they took control of, in the 19th century sense of the term anyway. Jews were emancipated; education emphasized, class distinctions became more fluid under the models of meritocracy as exemplified by the Army and the civil bureaucracy.
At the same time, of course the Empire was not a democracy, nor was it radically egalitarian; incorporation into the Napoleonic system was often negotiated by local elites who remained in charge locally.
So the system was pragmatic, focused on achieving results.
We'd need someone to carefully develop a TL which explains why and how the pattern of endless warfare changes to one of a truce that lasts; in that divergence itself there might be counterproductive elements, but there might also be a turn on Napoleon's part (or an eclipse of the Emperor himself as a figurehead--say he has a stroke or something like that--and his court is taken over) toward turning Imperial policy away from war and more toward "struggle by other means," specifically a program of rational development for the territories now controlled.
Development of the vast European territory need not follow the British laissez-faire model; indeed OTL few nations heeded the political gospel of the Smithian free traders, at least not when they had a choice. I will concede this much to these doctrines; the sort of top-down, dirigisme model of Continental development I vaguely imagine will interfere to an extent--but it will also assist and augment. The upshot might be overall less efficient development, but it would have some advantages as well--social conflict in the wake of transformative development might be better controlled and managed, for example. Workplace conditions might be notably better and safer than British factories. A continental Ministry of Development might interfere with the pure freedom of enterprise--but also support useful scientific research and engineering development programs, or provide capital for innovators who otherwise might not get it.
If there is any tendency for such an elaborated Imperial system to ossify or simply lag, this could be offset by observation of British successes and deliberate campaigns to catch up and surpass the rival power.
Also, consider that the British position interacts with the alternate Continental one--bluntly put, with whom will the British trade if not the Empire? This was part of their incentive to break that Empire up of course, but if we take it as proven that it lasts despite British schemes to destroy it, at some point relations have to normalize for British goods to find adequate markets. If the powers remain antagonistic, British growth will be stunted, even if they gain complete command of all overseas merchant trade. More likely, a symbiotic relationship would develop, or the Empire will assert itself enough to trade and maintain a powerful war fleet guaranteeing direct access to world markets. I envision a mixed situation where the Empire's global presence is less in proportion to population and general wealth than would be fair, but competition does exist even overseas; the Empire might have lost all overseas colonies in 1807 but would regain some at some point; Imperial merchants trade in American ports and probably have some presence in Asia. And while Britain's position outside Europe is glorious and profitable, it is also well understood in Britain that they are heavily dependent on trade with the Empire. That Empire would hardly be in the vulnerable position of Manchu China; its officials would be able to regulate the trade to be sure that not all advantage goes to the British, and there would be no question of Opium War type interventions.
So yes, I say yes even bearing in mind a long list of factors that might impede the Empire's progress. On the whole, even with liabilities, the assets would tend to prevail and a Napoleonic Empire of many generations duration would develop at least as well as OTL and thus hold its competitive position versus Britain.