Strongly recommend Gabapentin. It doesn't completely knock out your cat and as I understand it doesn't have the adverse effects your vet is probably worried about (notably reduced breathing). Instead it acts as an anti-anxiety drug, making them less anxious and stressed. Be warned that it can make them a bit loopy, although that shouldn't be an issue if your cat is confined to its carrier.
Also recommend asking for the private screening room at the TSA checkpoint. As others have said, the agents can have you wait there with the cat while they scan the carrier - that way you don't have to worry about the cat getting away. I'd also recommend a harness if possible, might make it a bit easier to keep hold of your kitty while you're going through all that.
If you haven't checked flights, I would start looking now. Nonstop would be ideal, but also note that airline policies on in-cabin pets vary quite a bit. Some are more strict on size/weight than others, some don't allow in premium cabins, etc.
Good luck! We're moving with ours from the US to Stockholm early next year and can't say I'm looking forward to it (cat is pretty chatty and has never flown before either). I'm starting the paperwork now, planning to utilize all the above and just hope it goes well...