Spending nights at two homes?

I am currently fostering a 4 month old puppy. It's so far going ok. She's settling in to my place and seems to be getting more comfortable.

My girlfriend lives about 15 minutes apart. We have different work/school schedules such that we spend the night depending on who's got the early morning or gonna have a late night or other things like that. She also has a cat at her place.

My question is does anyone have experience or advice about introducing/settling a puppy into two homes? On nights where I can sleep in the following morning (or more importantly go to class late), I would ideally like to bring the puppy over so that I'm not leaving her alone overnight and since I would have extra time to transport her back to my place.

Is it possible to introduce her to a secondary home and have her be comfortable in it? The organization I am fostering for told me to crate her so I've been thinking about wondering about a second crate at my girlfriend's place. I'm not 100% sure that there is enough space for a crate at her place (on account of the cat) so I'm thinking about if having her sleep in a crate at my place and not at my gfs would create confusion and potentially mess with the crate training.

Otherwise, I would obviously take the process slow to get her used to the second location. I'd intend to follow the same bathroom and feeding schedule in the new home. But otherwise I'm wondering if there's any major reasons this couldn't happen or anything really important we'd need to figure out before that.