Hey!
I find buns generally handle moving pretty well and maybe show signs of stress for a couple days if its their first or second time. I’ve moved my buns many times when I was a student and I always found the key to settling them in is familiarity over anything else.
I would put your buns cage and home base in the location where you want it to be permanently, so if you want to continue having him in the lounge just put it in there. Fill and surround it with all his fimilar belongings as much as possible. I would just allow him to explore as he pleases in the room that his base is in with the door shut if possible. You can keep him in his home base room for the first night or two as to not overwhelm him and get him to feel safe there before letting him explore the rest of the areas he will have access to.
It can really just depend on how confident he is. One of my boys I’ve had for over 5 years is a seasoned traveller and in our most recent couple moves we’ve just set up his area, popped him in it and left it open and the first thing he does is check out the house and binky everywhere. .. I think he understands that if his stuff moves to a new place thats now our territory so he’s excited to explore… But on the other hand, the first couple times he moved we kept him to his home based as he was very stressed by it all. So its really a matter of seeing if your bun is feeling stressed. If he is grooming, eating, flopping and otherwise unphased he’s probably ready to explore. If he is quite and withdrawn give him more time to settle in his home base.