I have moved rabbits on several occasions and not had any problem with it. Apparently the rabbits did not either. I try to move the bunnies several days to a week before or after the mover's big day, depending on the circumstances. I try to maintain their routine and diet regimen, as those are so important for giving a sense of orientation and reducing stress. I wonder if some of those stress supplements, like BlueSeals Snshine Plus, which I want to try in the near future-- my rabbits eat BlueSeal feeds and combinations there of. There are a number of supplements out there for stress, and maybe that's something you might want to look into.
Of course, I have more than one bunny, so the social fabric also tends to provide support to the rabbit in the new environment. With a single rabbit, I would spend plenty of extra time with him as he accustoms himself to the new location. If you have already located a reliable and trustworthy vet, you might want to ask that vet for tips on stress reduction for the rabbit.
I once had to keep two of my babies at the vet's, boarding, for a week. They seemed to do just fine there, as the vet was well-equipped to care for them. If you are contemplating this as part of your plan, make sure that the rabbit will not be boarded close to other animals that they are not comfortable with -- predators like cats and some large dogs. If boarding is an option for you, that might be worth considering for several days surrounding the official moving date.
I know a lot is said about rabbits finding travel and new situations stressful, but my rabbits seem to enjoy the opportunity to explore new spaces. The house rabbits we take with us on road trips, and they love each new hotel room we stay in. I am thinking that rabbits are like horses in that they are sensitive to, and read off, a lot of information on their owner's attitudes and concerns. I suspect that if you are anxious and stressed that could predispose the rabbits also to be anxious and distressed. When they see you being your comfortable usual self, that reduces stress and helps them be at ease and their usual selves.
Good luck with the move!
