I just came back from the DR flying JetBlue. I took all my gear (three kites and board) in a "Golf" bag.
I had the same concerns as you, they didn't even ask me what was in my bag - the only things they seemed to care about was the weight.
I just came back from Cabarete on JetBlue. I took all my stuff in a deceiver bag - the only thing they really care about is that is weighs less than 50lbs, but they are super, super, super strict about the weight.
Cabarete should be exactly what you want. I've travelled there many times on my own and always had a good time. It's very safe and you will meet tons of other kiters to hang out with.
If you want more info send me a pm - I'm in Cabarete right now.