I would probably grab an inexpensive infant seat to bring on the trip. If there is room on the outgoing plane to use it, awesome -- use it on the way out, in cars at your destination, on the way back. If there isn't, gate check it and then borrow a seat for the cars on the other end. On the way back, I'd bring the infant seat (even if it was gate-checked on the way out) and try to use it again for the return flight.
Provided there has been no visible damage, I would feel comfortable using a gate-checked seat on a plane and feel that it is safer than a lap baby.
I don't know that there's any way to guarantee the safety of a gate-checked seat in a car. It's good that you have a seat you can borrow on the other end either way. If your regular seat is nice enough that you'd not want to have to replace it, I'd get an inexpensive seat just for the trip.
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