I can't offer any direct insight into BTV airport or any of the three listed companies. But, I routinely have to deal with this whenever I'm renting a car while on work travel. When my travel agent makes my res for a rental, it gets sent with the rental company's loyalty number AND my work credit card for payment. BUT, at the rental company, in my loyalty profile, I ONLY have my personal (e.g. non-work credit card) in my profile, so the rental company uses my non-work credit card in my profile to hold the reservation. I ROUTINELY, have to go to the desk on arrival and say "here's my work card, please change." It's a time suck! But, they do it.