What can be done if the claimant was not injured but filed a bodily injury claim?

Relax and let your insurance co. handle it. That's why the law requires insurance.

You hit this driver from behind, so you would have to prove the claim was false. Did you have a set of excellent photos of the bumper showing absolutely no damage? You could file a small claims suit, but as a former small claims judge without written and photographic proof that there was no impact at all, I would dismiss your lawsuit.

You can not sue him. You were at fault in hitting his vehicle. You should not have been driving so closely behind him so as to hit his vehicle, even if he stopped suddenly. You may have been going slowly, but not only 1 mph. Your insurance company would not have paid for a bumper replacement unless they felt there was sufficient damage as to require it. Even at a very slow impact speed it is possible to suffer some injuries. So it would be very difficult to prove that he has made a false claim. I doubt any attorney would be willing to take your case. If he was not injured, your insurance company will take that position. As a practical matter, this is just one of the the various insults people have to face in life that they can not really do anything about.