The solution here could not be simpler,
Just hire a car in the UK (pick it up and drop it off anywhere you want), and
Just hire a car in Europe (pick it up and drop it off anywhere you want).
(It's completely trivial to transfer between Folkestone and Calais or whatever, or between some UK airport and Lyon or the like.)
It's utterly commonplace to do this. You'd be one of a trillion tourists doing it this year.
The idea of using one car for both UK/Europe is a total non-starter, just forget it.
To go against what you say in your question:
- As many have said, save a massive amount of money by making a loop rather than a one-way rental.
https://travel.stackexchange.com/a/38154/19233
One-way car rental in Europe is fantastic, but it costs a lot.
- Whatever you do, in the name of goodness avoid the "living hell" major airports (all London airports, Paris). Fly to Lyon or Munich or the like, and the quality of your life/holiday is boosted to a different level.
One of the great joys in life is motoring around the Continent. Enjoy!
Just for the record, note that it is
#no big deal to take a UK rental car to the Continent
if for some reason you want to do that.
Funnily enough here's Sixt promoting the concept for a day trip:
https://www.sixt.co.uk/one-way-car-hire/london-to-paris/
(Interestingly though, Sixt state they do not have one-way rentals to the Continent.)
You would never do this for a longer tour, simply rent a car on each landmass. Enjoy!