OK, I would have a problem with a TSA agent telling me I couldn't take a car seat to the gate even if my dc didn't have a ticket. I don't know where or why they would have the authority to tell you to check it with baggage.

Airlines allow car seats to be gate checked all the time. For one thing, most airlines will allow you to use a car seat for a non-ticketed baby if there is room on the plane. Unlikely on a flight to Orlando, but you never know.
OP, I honestly think you take a risk either way. If you baggage check, it can be tossed around and have heavy luggage put on top of it. If you gate check it, I've
heard of things being tossed down from the jetway to tarmac. I've never
seen that happen though, but that doesn't mean it's not true. One other concern mentioned about baggage checking a car seat is that it is more likely to be misrouted or lost. I have heard really terrible stories about the loaner car seats people have been given by the airline to use when this happened. Airlines also do consider car seats to be "fragile" and will not cover the cost of it, if it's broken. OTOH I would not want to risk relying on a car rental agency for a rental seat for several reasons either.
We have checked our ds car seat on one trip, in a car seat bag, with pillows for extra padding, and it came thru both ways just fine. No indication that it had been mishandled in any way. We did
not need the car seat at the airport though since we were taking DME to our hotel so I wasn't as concerned about it getting lost. We used DME and then rented a car from the agency at the Dolphin resort (near the Boardwalk, where we were staying). It's still not something I really recommend doing. Up until ds was four, we used the car seat on board the plane because not only was he safer in it, but also it assured us that the car seat would not be damaged or lost enroute.