Where are you flying to and where from? I took my Betta on a 6 hour plane ride back to Oregon from Hawaii and he was just fine with me on the plane. If you just hold him in you lab under the blanket to keep him and check on him.
You should call TSA first and clear it. That's what I did and I didn't have any problems getting him through security with me. You could keep him in the betta cup that you bought him in with water from his tank. It just has to be a clear, some-what small plastic container. I took a small rubbermaid one and put a hole in the top so he could breath.
And just for clarification, I didn't have a problem getting him on the plane to go back to Oregon. I could have even brought him back if I had known that it's really not that hard to sneak him off the plane back into Hawaii, they don't check your carry-ons.
But I didn't have a problem at all, just call TSA and ask. They will just do a visual check because they are aware a Betta needs to be in treated water. Sometimes their do a chem test. But yes, they can go on the cabin with you.