Please dont rent a carseat - they are scary!
Besides, on the plane your little one should be riding in their carseat anyway, its much safer.
If they are going to be a 'lap child' then gate check your carseat (do NOT put it in the belly of the plane!), you can get one of those rolling cart things to carry it through the airport.
I've traveled with my ds, his carseat, a rolling suitcase (which I did not check in), and a carry on bag - and I did just fine - I put...