Personal opinion, but I would never use a carseat from the rental place. We've been warned for decades to not use a carseat after even a small accident and since it's your child's safety, well, I never would use a car seat that I didn't personally know the history of.
When we travelled with the kids we simply brought the car seat on to the plane with us.....they were more comfortable in their own car seat than sitting in that big ole plane seat (if only it felt big to us!!). If your child is young enough that you didn't buy a plane ticket for her, take the car seat along anyway.....they will stow it in the cargo hold for you without charge. Now it's been quite a few years since I've had children that young, but even when we intended them as "lap babies" we brought the car seat right to the plane door....IF there are seats available we were always given a seat to put the car seat and baby in. Now I realize it's been years and they may have finally caught on, lol...and that flights are now regularly over booked so it may not work for you, but since they'll take the car seat to cargo hold right from the door of the plane, you have nothing to lose trying, lol.