We have done 5 plane trips since my son was born. We have done both.
TSA approved is basically a car seat with a back if memory serves me.
How is your 3 year old.. Active, or calm? Active and does not sit well, you might want to put the car seat on the plane. Calm child, gate check it.
We have a bag that we got with rollers to put the car seat in. It was about $50.00.
My sons first trip, he was on my lap. 2,3 and 1/5/2 of 4th trip (on the way there) we did the car seat, On the way home and this last trip we did no car seat.
It is a pain to bring the seat on and off the plane, but the work is sometimes worth it depending on the kid. For my son it was worth it.
This last trip, we did not bring his seat with us, just my daughters. We did not have a plane ticket for her, so she sat on my lap.
Either way, put it on, or gate check it.. do not put it in as luggage.
Good luck and happy travels.