i think once he’s on limited pellets, the hay intake should increase. most bunnies love their veggies and will actually pick veggies over pellets, altho you have the few that would only eat pellets if given the choice. you might want to feed them at separate times, like veggies in the morning and then he can graze on them all day, and pellets around dinnertime.
i would give him one handful of hay at a time. bunnies tend to eat more when it’s replenished frequently. so check the hay a couple times a day and add a bit more each time. a hay rack is nice b/c the hay doesn’t go to waste, but i know with some buns it’s necessary to have hay in the litterbox.
can he come and go from the cage as he pleases, or does he have to be lifted out? you might want to try expanding his territory by attaching an xpen so he can feel secure in a larger area. he might not be an explorer.
what kind of “outside cage” excitement does he have? put some things he likes outside of the cage so he has to leave to get to them. you could try cardboard boxes, phone books, other toys, tunnels, or even the new BB MazeHaven. the more places he is made to feel comfortable in, the more likely he’d be to come out.