The moment Britain started taking huge swathes of territory, it would become uncompetitive and collapse. It is no coincidence that the rapid expansion of the empire coincides with the rapid relative decline of Britain's industry - the taxes needed to fund such a empire need to come from somewhere.
Napoleon got to Moscow and realised he was ridiculously overextended, such over-extension lead to the Louisiana purchase and the numerous truces agreed with the coalitions.
Alternative history is one thing, fantasy another.
I think you are a bit confused with the Napoleonic wars as the Louisiana purchase happened in 1803 and the Russian campaign occurred in 1812. As such it is ridulous to say that the second led to the first. Napoleon sold Louisiana because he needed money for his campaigns and avoid conflict with the United States of America over the region and a consequent alliance (even if I think that was unlikely) of the US with Great Britain.