When your child is young enough to be carried on board in their bucket, buy them an extra coach seat and strap the bucket into the seat with the kid in it -- The Chico Keyfit 30 system works incredibly well for this, and for rental cars at the far end (check / gate check the base the seat clips into). The lap belt (ask for an extender) goes over the top of the seat under two little clip arms that holds it snugly. Baby is happy because they're in their familiar bucket.

When they grow out of the bucket and can fly forward-facing, buy a CARES harness and check their car seat before boarding. Flying gets SO MUCH EASIER when you can make that transition, as you're only carrying on the little red bag with the CARES in it. For those who don't know it, CARES harnesses are basically FAA-approved shoulder straps made out of seatbelt material that go around the seat and serve as a five point harness using the existing plane lap belt. Occasionally you'll get a new FA who hasn't seen one, but they all come around quickly.