Yes, it is an interesting scenario as it can lead to any number of different conclusions.

However, would Britain still be at war by 1812?

Britain had to maintain at least 12 SoL off Cadiz alone and then still blockade Ferrol, Vigo and Cartagena. Its export had fallen to 37 ½ million in 1808 and without those South American markets how would it rise? (rose to 47 ¼ million in 1809 and 48 ½ million in 1810). There was industrial unrest in the country due to the lack of trade which had it continued to grow could have paralysed the country with unrest. . The poor harvests of 1809/1810 caused hardships and when Napoleon sold corn to Britain he did so to drain its gold reserves. The same reason he allowed trade licences but demanded payment in gold.

Britain was planning to attack the Spanish American colonies. This might have entangle it in the same kind of fighting that Napoleon found in Spain and even with better generals than Whitlocke perhaps with the same drain on manpower and resources. Britain still had to take the West Indies possessions of France and Denmark (in 1809/1810) and those of the Netherlands East Indies (in 1811).

At some point before 1812, Britain may have had to make peace with the French Empire.

Whether or not Europe would have remained long under Napoleon's rule would depend upon what Napoleon did any peace.

Good hunting

Ron