HELP NEEDED AND APPRECIATED: Is it dangerous/bad to start bringing my rabbit (who is not used to traveling) to stay overnight at my girlfriend's apartment with me a couple nights a week?

So Rocky is 5.5 years old, and has only lived/spent time in 2 different places: my law school house and my current apartment.

In both residences, he lives/lived in a large pen (here it is).

That being said, my new girlfriend loves Rocky and we'd both like it if I was able to come stay the night at her apartment (less than 5 minutes from my own apartment) without having to be stuck at hanging out/playing with Rocky at my place until like 9-10pm.

As such, we are considering the idea of making a small pen/hutch for Rocky at her place, and me just bringing him over on nights I want to stay with her at her place.

The issue is that rocky is not a well-traveled rabbit, and he tends to not like to leave his comfort zone.

He loves my apartment, but he doesn't even like going off the living room carpet and when I moved from my old house to my current apartment, he wouldn't even come out of his litter box in the new pen for almost a day. I almost thought he was going to die.

Does anyone here have any suggestions/warnings/advice as to how to get Rocky to accept this kind of travel a couple nights a week, or do you think it's too dangerous to his health?

Thanks so much for any and all advice/help!!