I always suspected that, once France had lost Hatii, making an empire in the Americas a logistical nightmare, Napoleone saw a different, strategic value in Louisiana: not keeping it, but rather selling it, specifically to the U.S., figuring an enlarged U.S. becomes a problem for Britain and a hinderance to their efforts to oppose him.
Might have worked too, had Napoleone made millitary advisors (officers and non-coms from the French Army, at the time the finest in the world) part of the deal.
It very well could have saved him from the Peninsular War, or,at the very least, blunted the British efforts there, as a U.S. with a well armed, well trained, and most importantly, well organized army could have opted for war with Britain instead of enacting the disasterous Embargo Act of 1807.
Contingent, of course, on Jefferson building up the U.S. Navy and letting the better trained and organized army plan and conduct the war, rather than having Jefferson and his cabinet and the congress muck it up.
Consider an LP that includes French millitary advisors and Jefferson and the Republicans in congress build up the navy "For the purpose of securing neutrality."
The LP goes down as it did OTL on May 2, 1803 (Treaty signed) and the French advisors are already in Louisiana (sent there when it becomes apparent the accord will be reached) or a short time later. The advisors could teach and train the American officers and non-coms for better than two years before the start of the War of The Third Coalition, plenty of time for the French advisors to see what a mess the U.S. millitary establishment was (and organizational nightmare) and rectify it as well as have training manuals translated and implemented to all officers, non-coms and enlistees.
Starting in 1803, the U.S. army is training in the ways of the Grand Armee, with the capable promoted, the patronage commissions and others unfit for command sacked and replaced by men actually capable and worthy of the positions.
Meanwhile, the Navy keeps building; no need to bring the French into the naval/marine corps part of this. The U.S. produced the best ships, officers, crews and marines of the era...they just didn't produce enough of them, (or some would say of the ships, not enough OR big enough) as would be demonstrated in 1812.
So in TTL, you have a larger U.S. Navy and better organized and trained U.S. Army. The British, as in OTL, impress U.S. sailors on the high seas, pass the Orders in Council, blockading Europe and then the final straw The Chesapeake-Leopard Incident on June 22, 1807.
Rather than passing the sub-moronic Embargo Act of 1807 as he did in OTL, Jefferson opts for war.
Jefferson once said that conquering Canada would be "a simple act of marching". The Army, far too diciplined from two years of direct Grand Armee training and conditioning, and a further three years of continuing the practice and implementation of such to ever consider harbor such a foolishness mindset, wouldn't make such a mistake.
They'd inform Jefferson of what they needed and how much. They'd be organized and efficient. They'd have battle experience from frontier clashes with the indians and the Navy and Marine Corps. would have experience from The First Barbary War. They'd have confidence, but not a sense of invincibility. They'd prepare for an inavasion of Canada much more carefully and in much more detail than OTL's WO1812.
In the meantime, the congress raises neccessary funds to construct more ships, raise sufficient soldiers, sailors and marines, take measures to make sure they are well equipped and provisioned and bolster the navy as necessary by issuing letters of mark.
Figure by late spring early summer 1808, juuuust after the first British troops have landed in Iberia and committed to battle there, the army and navy are ready and perhaps congress has secured some naval assistance from France. (Never hurts to have a few extra ships to assist on the other side of the Atlantic, in the Med and Carribbean.)
Congress declares war against GB and the U.S. invades Canada.
Now, even a better trained, diciplined, equiped, provisioned, funded and lead U.S. Army may or may not succede totally in it's war aims. Same for a larger, better funded navy. There are no certainties in war, BUT, I'd have to believe they'd certainly outperform the mess that went to war in 1812.
I don't know what the outcome of "The War of 1808"/"Anglo-American War" or whatever it would end up being called would be. What I can be fairly certain of, is that it would definately disrupt the British war on the Iberian peninsula and that would have ultimately served Napoleone far better than his simple land sale did OTL.
So, Naps, and the U.S., would probably have been better served to have millitary advisors as part of the deal.
He might just have succeded in accomplishing what I suspect was his prime motive in selling it to the U.S. all along.