I have only seen them weigh a pet once at the ticket counter and that was because it looked pretty big in the carrier. However, I have seen several agents at the ticket counter and the gate have people put their carrier on the counter to make sure the animal can easily move around and determine they aren't too big for their carrier. I don't spend a whole lot of time at the ticket counter so I'm more familiar with the gate agent's activity, but I have seen it happen many times.
I also echo not to take the kitty to the lavatory, I think it would just frighten it and you would probably have a heck of a time getting it back in the carrier. Friendly as a kitty might be in the comfort of your home, I've seen them turn into vicious scratching and biting animals when they're in unfamiliar and loud surroundings.
Good luck, I hope it all works out for you.