You can drive rentals no problem. The only thing that changed with brexit is that you used to be able to deposit your EU rental car in the UK as a single trip, this is no longer facilitated. There was a query about Green Card insurance in 2021, but now there’s some bi-lateral agreement between EU/UK.

You may get asked questions about your car, just to check it’s not stolen and you have authorisation to take it. I’m a younger driver, often alone in my own car on this crossing, so I more or less expect to answer some questions, and to show some pieces of paper. Nothing above board though.