What if: French invasion of the United Kingdom (1803)

  No problem.  As you say I've studied this period a lot and its easy to miss something.  Done it often enough myself.

  To be honest no idea.  He probably didn't have much himself as he seems, especially in the latter years to have been conquer the opponents then often a fairly draconian peace.  Would expect a long occupation and quite a harsh one as he really hated Britain's continued opposition to him.  It might also mean he starts thinking again about French empire outside Europe so you could see him, if he managed to do it seeking to reclaim Louisiana from the US.  Which probably wouldn't go down well with them.

  Well this was a couple of years before he put his brothers on assorted thrones [Joseph Spain, Louis The Netherlands and Jerome Westphalia so he hasn't done that yet but they are available.  Since he's already moved back towards Catholicism and had the pope present at his crowning as emperor I suspect that putting the Stuart claimant on the throne is the most likely.  As well as no doubt keeping a large garrison in Britain and probably taking massive taxes and a number of colonial locations from Britain.  Don't know if he would try 'reclaiming' Canada although between the Anglo settlers and the fact the French ones were from the former Bourbon kingdom and socially pretty conservative I suspect that would be unpopular.

  I doubt at this stage he would try ruling it directly as it would be potentially very explosive, both in Britain and the rest of the continent.  Deposing an established dynasty would upset the conservative states as it did OTL but replacing one with another with, arguably a stronger claim would be less explosive.  Also with Britain defeated and hence no naval rival you could have a markedly more relaxed stance with regards to the other states so you quite probably wouldn't see the Bourbons deposed in Spain.  Could well see Westphalia established as Prussia is still largely undefeated and you might just see it, Austria and Russia combine in one alliance because of the threat from Napoleon. 

  He might try splitting it up, most likely removing Ireland, that would weaken any restored Stuart monarchy, whereas since that dynasty had converted to Catholicism Ireland would probably be its strongest supporters so that would seem unwise to me.  [Mind you that might not stop Napoleon. ]

  Possibly sell the Anglo areas, although I think keeping Quebec is almost certain to lead to later conflicts with the US, which would become increasing difficult to sustain over coming decades, although I doubt Napoleon would look that far ahead.

  Very likely.  Suspect he thought conquering Britain would solve his main problem but not really much beyond it.

  Well the Stuarts were originally Scottish and ruled it for quite a while before inheriting the English throne through marriage so it would be difficult to find an alternative Scottish dynasty, at least with an historical claim on the throne. 

  Ireland never really had an Irish monarchy, although in the distant past assorted claimants to the title of High King of Ireland as power switched between the assorted kingdoms, but their not been in existence for so long it would be difficult to find a claiment.  Also as I said the Irish, being mainly Catholics are the most likely substantial group to actually be loyal to a Stuart monarchy so I think it would be best to maintain the link.

  Any way you would need a fair number of French troops to keep the English and Scottish down.