My rabbits have all done this until they were about... 4 or 5 years old. No matter how many nice toys, run around time, or cuddles I gave them. It's not about the paint per say. I bet once this bunny chews through the paint, he'll continue on through the wood just like mine have all done. I assume it's just part of their natural chewing and digging burrowing instincts.
Paint isn't good for them to eat obviously. My rabbits have never had any health problems related to eating paint, but I suspect that if the paint had lead in it, that would be a different story.
You want to deter the rabbit from chewing on the walls. It takes time to train them though. You can't explain "no" to a rabbit, you can only deter them from the bad stuff and encourage the good stuff.
In my experience, the bitter sprays don’t work. NONE of them deterred my bunns from chewing the baseboards of the walls. In fact, I think they took a liking to a few of them!
I even tried using the juice of a jalapeno once on the walls thinking the hotness would stop my rabbits. Apparently, my rabbit Owen LOVES jalapenos. He took one taste of it & chewed with more enthusiasm than ever before. I think he thought it was a treat or something. Go figure.
The best way to deter a bunny from chewing on something is to make it distasteful before they even get near to chewing it i.e. put something smelly on it. Rabbits have sensitive noses. They don't like the smell of chemically smelling perfumes and colognes etc. I sprayed some cheap icky cologne on the baseboards, and my rabbit would smell it, huff defiantly and hop away. NO MORE CHEWING!!! This has been the most effective method I’ve found for stopping them from chewing on the walls. You have to keep reapplying though.
Another thing to deter chewing is to spray them with water while in the act just like you would do with a cat. You have to spray them WHILE they are in the act though, otherwise they won't understand. Over time, my rabbits understood that they shouldn’t chew on the walls WHEN I WAS LOOKING. LOL. They’d see me come with the spray bottle & they’d stop and act all innocent. So cute =) It takes time.
This could also be a sign that your bunny needs more socializing. Rabbits are very social animals. Your bunny needs time when he or she can hop around in the places where the humans walk to interact with you.
I've been a bunny mommy for over 16yrs