I'll join the crowd and say "no way!" to renting a car seat from a rental car company.
On the other hand, there are baby equipment rental companies in many locations and if you feel you really can't bring your own seat, look into what is available in the area you are going to. I used to be a contractor for one in SW Florida and we had gorgeous, 100% clean, nearly brand new car seats (not the cheapest we could buy, but about medium) and the owner allowed us to buy new seats practically whenever we felt the need. I always felt it was a very good deal. But then again, depending on how long you are going to be there, purchasing a safe but less expensive seat just for the trip might be just as economical.
Depending on the age of your child, I would recommend bringing the Britax on the plane. I always loved my children to be comfy for naps on planes and safely buckled in in their seats (in case of turbulence). For children who are stressed by the idea of being on a plane, I feel that the comfort of having their own seat there might help a teeny bit.
eta: btw, once I was bringing a couple of rental car seats to a car rental agency for customers who were flying in, and I saw the incredibly crappy seats that the rental agency had, stacked in their garden shed out back. They were absolutely horrendous. Should have been thrown out. It was criminal, I felt.