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:

1. 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

![enter image description here][2]

One-way car rental in Europe is fantastic, but it costs a lot.

2. 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!

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Just for the record, note that it is:

no big deal to take a UK rental car to the Continent for, say, a day trip

... if for some reason you want to do that.

Funnily enough here's Sixt *promoting* the concept of a day trip!

https://www.sixt.co.uk/one-way-car-hire/london-to-paris/

(Interestingly, 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!

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Ahh!  I just remembered, I once rented a car in France (near Switzerland), drove it to the UK, collected some stuff, and drove back to the same place in France over a couple days.  If I'm not mistaken, there was not even any extra paperwork or costs for that. And I'm pretty sure I would have rented a car in the South of England on occasion, to drive over to France for a day trip. So it's no big deal to do that.

(But again you would never, ever, take a long tour with a wrong-side car; it's one train stop or a dirt-cheap flight between the two landmasses. It's a non-starter.)

  [2]: https://i.sstatic.net/h8b5a.jpg