We've got EU passports for our cats, but it took at least a few days for them to be sent off from and then returned to our vet's office. We didn't do it to circumvent anything, just wanted to make sure they maintain current documentation for any future travel (we're not in the UK permanently, only for work), and it was the easiest way now that we're in the UK, although I have their US paperwork and lab results filed away, too.
I didn't look into it, but if it was even remotely easy to get an EU passport from the US, it seems like that's what we'd all be doing. It would have been way easier to have the little passport for them than worry endlessly about the EU998 paperwork and the ink color, date format and the vet stamp, USDA certification and all of that. But it didn't seem to be an option.
My cats are older, and every minute shaved off an already arduous journey was worth it for us. It's definitely expensive, but so was everything else for this move and they're our family.