Both of my cats are careful to keep their back claws trimmed, but their front ones long and sharp. They do this by carefully biting their back claws so that they are shorter and less sharp. Why do they do this? Wouldn’t it be to their advantage to have sharp claws in the back as well? Is this some instinctive action?
They use their back claws to scratch their itches. If the back claws were razor sharp, they would lacerate their own skin. The back feet normally have thicker, more blunt claws, as they use the back feet for jumping and if they got stuck on a surface during a jump, bad things would happen.
I haven’t actually seen a cat really trying to “trim” the back claws in the way you describe, though, since they are usually slightly blunt compared to the front. I have seen cats groom their feet excessively if they come in contact with undesirable substances such as floor cleaners or kitty litter clumps. I’ve seen cats chew at their toes when they have allergy issues. Dogs, too, for that matter.
My cat also bites at his back claws. In his case, I think it’s because the outer claw sheath doesn’t come off by itself on his back feet. (Cats shed the outer layer of claw periodically, somewhat like a lizard shedding the old skin).
If he will let you clip his claws, you might try trimming his back claws short and see what he does. Maybe he will then quit biting.