To be taken with a grain of salt as I:
- have no idea how comfortable you are with driving long distances (disclosure: I am fine with ~1000 km a day, depending on roads and traffic)
- have no experience driving on the left side of the road, whether in an LHD or RHD car
Asides from that, based on my personal experience, having a car that you are used to is a boon, whereas having to go from automatic to manual is an annoyance at best. The latter is exacerbated when you are driving on unfamiliar roads (even if it happens to be the area in which you grew up or have lived for a long time, road layouts may have changed beyond recognition over the last 20 years).
The other thing to consider is the financial aspect. In the bring-your-own-car scenario, cost drivers are fuel for the trip, wear and tear for the car (with the slight risk of your car breaking down or you having an accident on the way – of which I just had my share), road toll, Eurotunnel or ferry to get to the other side, as well as possibly some place to stop for the night. For flying, you need to get to the airport, pay for the flight and possibly for extra baggage (if you are carrying more than the standard allowance), then rent a car at your destination. Either way, you will do the whole thing in reverse on your way back.
So, my personal inclination would be to take my own car rather than fly and rent one anytime, unless there is an ocean in between (i.e. significantly more water than the Channel). But your mileage may vary.