I think it’s great that you’re being observant and considerate to your new rabbit’s potential needs — seeing what you believe is depressive behavior (i.e. destruction when previously was not destructive, hiding) can be distressing to see.
For one, I think it’s important to make sure he is acclimated to your home environment first. Bonding can be stressful, and if you chose to bond through a rescue, you want to make sure the return home is one that’s comfortable.
I am not personally experienced with bonding, but I think generally rabbits will still “remember’ their owners, and if not, they will because you are the source of food and that’s a might strong relationship-building tool!
Whether it’s a solution or not, I’m not sure, but based on what you’re saying, companionship other than human does seem to be a likeable route for your rabbit.
I will add, in regards to the cats that roam when you leave the door open (?), I would make sure to monitor at all times, especially the cats that bolt. Wick and I temporarily stayed at a house with a cat and the cat is a huge coward— never hunts, is scared of rats, is extremely lazy one top of that… not a predator in the least. He ran every time he saw Wick or got near him, or when Wick moved.. but after some time, the cat got more acclimated and started hissing, stalking, and looking to pounce at Wick. So keep an eye out for that. A playful bat from a cat can be very dangerous for a rabbit, not to mention traumatic. Wick was chronically stressed from this and it really showed in his affect and energy when he too began to realize a cat was stalking him and trying to get him.
The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.