Guys, you’re all getting bent out of shape for nothing over this TravelBank stuff. Book a ticket with whatever credit card you want on United. Wait 10 minutes for the ticket to confirm. Then, add a pet for $125 each way and pay with your Amex. Wait an hour or so. Cancel ticket (pet charges cancel automatically at the same time. Watch your Amex for a credit a day or two later.
NB that you can’t book a pet on all flights. Try routes with ERJ 145s since they don’t have a first class cabin. I find that works best.