Heyo! Welcome to the forum! It’s good that you know his past diet, as having that knowledge gives you a good foundation of what to expect at the vet visit.
Hay is the exception to the “slow introduction” rule for food I think— rabbit diets should be 80-90% hay, so do not shy from offering it.
The quantity of food items will depend on his age. You say “adult bun”, and considering he’s 15lb (potentially overweight due to past diet; your vet can determine that), he is probably well at the age where he should have limited pellets and unlimited timothy/orchard/oat hays. I am not experienced with larger rabbits, but I believe the ratio is 1/4 cup of pellets per 2lbs? If you have Oxbow pellets, the feeding instructions on the back are proper.
Have you tried offering hay yet? Ways to entice hay consumption is 1) offering fresh hay, 2) scheduling when you present non-hay items, so when he’s hungry, he only has hay to eat for long periods of time, and 3) finding his favorite hay type (typically rabbits really like orchard).
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.