Ack! That carpet has to be pretty darn gross by this point!
We try to enforce a no-shoes-inside rule, but still wind up sweeping our bamboo floor (not recommended) almost constantly when camping. I cringe to think about what kind of an archaeological dig that would be if the original shag had remained! Not to mention the moisture-retention factor, and haven for pests.
If you're interested in the retro look, may I recommend the following sites:
https://retrorenovation.com
https://heffrons.com
If the entire bathroom is out, I'd use old-school linoleum for the floor. Not the tiles, but a single sheet. You want to be very careful to seal off the floor, otherwise you'll be dealing with the dreaded bathroom floor rot that plagues so many older Airstreams. Maybe use some heavy duty plastic, like tile setters use in showers, that wraps up the sides (under the interior skin).
My bottom line: go with some sort of resilient flooring, and maybe use some shag throw rugs to get that cool 70s party-pad look. You can always shake them out, or throw them away when they get gross!