I don’t think it’s claustrophobia. I imagine if it’s more “cave” like he’s more apt to not see what’s going on and just hibernate. I think a more open cage with a place to get away is a better idea too. My rabbits are in puppy pens with a wicker tent they lay under and they like this setting.

I do have a pair in a 3 story condo that is open all around and they have a kitty condo they like to get into and onto but they don’t like to be bothered at all.

I would consider an x-pen for your rabbit for more space.

My male was in unknown circumstances and then in a tiny cage in the local humane society for 6 months with limited interaction. I built him a nice huuuuuuge open x-pen type cage (took up 80% of our spare bedroom), he freaked out when I locked him in it. Banged against the walls, jumped into the roof, was digging, running in circles and pretty much having a panic attack at being locked in. Open the door, he was fine, he would even go in on his own and play in it. Built him a different style cage, same thing when the door closed. I ended up having to just leave his cage open 24/7 and scrap the idea of caging him, to avoid injury. He now gets locked in a 3-room area that is bunny-proofed if we have to go out/at night while we’re asleep. So I would say yes rabbits can be claustrophobic. Mine certainly has panic attacks at being closed into any area.

I don’t think it’s exactly claustrophobia so much as really deeply ingrained preferences, especially the rescued rabbits who were in situations that were hard on them. Sammy will hang out for hours in her cage if the door’s open also, but chews if I close it. Sometimes.
I think it may be more the need to have the best available access to seeing what is going on (self-protection) and have options for dealing with it. A cave limits vision, a closed door limits freedom of flight.
bunnies are very unique in their individual natures to begin with, but add on responses to adverse conditions, and you just have a bunny that may need his environment adapted to his quirks. I don’t have any platforms for Sammy anymore because she gives every indication of being afraid of heights. But who knows for sure?
What’s so nice is that you are making a big effort in arranging his habitat to suit him, even if it means going back to an older version and abandoning the lovely new hut you got.

I wouldn’t do a light inside. Something he might be able to do (if he’s handy) is actually create a small square “skylight” (maybe even some side windows), and then finish it with some wire mesh. That would also help with ventilation. Another member (Mocha) has a wooden cabinet/hutch outfitted with windows.

You could try to create a positive association with the cage. Of course, you will have to have both cages set up for a while, which might not be doable space-wise for you… for such tiny creatures, they take up a lot of room! ;P
With some patience, you can build up his confidence. If it were me, I would feed him only in the hutch. That means taking him out of his usual cage and putting him in the hutch to eat and then removing him when he’s done. You can also do this through out the day with a few treats. Start with smaller times… 1 or 2 minutes and then work your way up.
It could entirely be that he’s just not comfortable with the hutch and not being able to look around, but rabbits [in the wild] DO burrow.. It’s funny because my rabbits rarely lounge out in the open! They prefer their little caves! Hehe

Is the hutch on ground level? Is the cage on ground level? It could also be a difference of perspective. If the hutch is 3 feet or so in the air, that visual might be frightening for him. While some rabbits do jump and climb, by nature, they are ground dwellers. The only reason why outdoor hutches are 3 feet in the air is to try and prevent attacks from predators, and allow a space for droppings to fall. This, of course, isn’t an issue in your home. Does the hutch have removal legs?