Hi, A. -- Search Mamasource archives under "Travel" for a lot of tips. Several moms have posted the same kinds of questions in recent months as summer has approached and you'll find loads of help there.
We've traveled to England annually with our daughter, now 8, since she was six months old. Here are two basics related to your post:
1. Do bring her own car seat. Hassle to carry? Oh, yes, a big hassle! But she will be more used to it; you KNOW it is safe and has not previously been in an accident (how can you know the rental company's seat hasn't been in one? Once a seat's in an accident, here at least, you're advised to get rid of it and replace it even if it appears perfect); and you know it's the right seat for her. I've had friends go with the rental car co. providing a child seat and found the provided seat too loose, hard to install or tighten, or the wrong type (company provided a booster for an infant, etc.).
2. Do buy her her own airline seat both ways and use her car seat in it; she is likelier to stay put because she's used to staying put when in her car seat. It would be ideal if she slept, of course, but there is zero guarantee of that, and the time changes, excitement, etc. mean she probably won't sleep much; being in her familiar car seat may help her get more rest. If she's on your laps, you and your husband will arrive in France exhausted and irritable from having her on top of you for hour after hour, whether she's sleeping on you or wiggling and fussing. Also there is a safety issue; if there is serious turbulence, you cannot necessarily hold onto her. In other words, I would give her the same safety level on a plane as you would give her in your own car.
One more tip -- if you do get her an airline seat, check that your carseat is "approved for aircraft use." Should be in the manual for the seat. Then call the airline and double-check with them. Not all seats fit on all aircraft or can be secured correctly with the aircraft seatbelt, but many, probably most, can.
Have fun in France! Self-catering accommodation is the way to go with younger kids -- we do it every time in England. Let us all know how it goes.