Hello Sam,
we went to Florida last October I took our own seat for all of the above reasons.
What I did find useful was,at the airport I placed the seat in a clear plastic bag with label tied at top. For security reasons the airport security could see what it was and there was no problems. I was also allowed to book it in and it was ok and undamaged at the other end.
Have a great time, I'm envious we don't go till October!!!.