Can you attach an x-pen to his cage in the living room, so that he can get his exercise there and not have to be moved? That still won’t give him as much room as having a whole room to run around in, but that would give him more exercise and you could take him to the other room once a day instead of multiple times. A rabbit really needs as much out of cage time for exercise and play as possible. Fifteen or so minutes a couple of times a day isn’t much time for exercise.
Most bunnies do dislike being held and carried and you do have to be careful that they don’t struggle and injure themselves. If there isn’t room to set up an x-pen next to the cage, maybe you could transport him in a carrier. The top loading carriers are a lot easier for rabbits. If you have the carrier near him, you could quickly pick him up and put him in and take him to the other room for exercise.
You didn’t say how old he is now, but if he is old enough to eat greens, which are usually slowly started around 12 weeks, you could try feeding him some favorite greens or a little bit of fruit as you try to groom him.
Has he been neutered? If not, and he is old enough to be neutered, that may calm him down a little so that you can more easily clip his nails or pick him up, but even spayed and neutered rabbits are going to resist those things. Some people take their rabbits to a vet for nail trims. There are some rabbit rescues that may offer nail trims too.