Right. Print a card with your dog's picture on it that has a little wheelchair symbol and says something like "Texas K9 Services - Cerification of service dog status - TX State License 5667-233A" and make it look nice and official, and hang it in a badge holder on your dogs neck. Also carry a copy of the ADA with you.
They can NOT ask you about your 'disability'. If they do, remind them what they are asking you is illegal under the ADA and whip out your copy. They'll STFU right there.
You can't just PLAY the part, you have to ACT the part. Like, mention that you may appear to be sleeping and if you are dificult to wake, not to worry or panic as long as the flight crew can see you are breathing normally.
I had airport police come up to me in Charlotte and tell me my dog wasn't allowed in the terminal once. A good glare and a 'not every disability requires a wheelchair' and I recieved profuse apologies.
It's risky, but IMHO it beats having your dogs crate accidently shipped to Beijing or getting dropped onto the tarmac and forgotten....