Those days are behind us now, but here is what we did. We brought our own seats on the airplane. They were FAA approved. Yes a pain to carry, but it was safest for our kids in them anyway. I also did not feel safe checking them (unless you have an original box) as I am sure you have seen how awful some on the baggage handlers manhandle luggage and we had nice carseats! I did not want their safety compromised.
We only rented a car seat once. And honestly we hated it. More importantly, my then toddler HATED the seat and did not want to use it. She found it uncomfortable. Trying to reason with her over this was not an enjoyable part of traveling. I would way rather lug my own car seat then deal with that!
And for the record there was many (too many to count times) we traveled with 2 kids in car seats (both on the plane, then both installed into the rental car). Good luck!