It might not be enough to exonerate him, but he doesn't have any other choice.
Point 1: British law in taking the prints
I can't remember where they find the fingerprint, but from the question I'm guessing it's an unidentified print in the victim's house.
The police had no reason to take his fingerprints before then because a. whilst he was a suspect, they had zero evidence against him, and b. they had found no fingerprints before then to identify his against.
This means they had no justification to take his prints before they found the print in the house.
Point 2: Only removing his right hand
When Luther phones Ken Barnaby, he knows from the phone call only that the police have found a single fingerprint, and that the police will be coming the next day to take his fingerprints to compare to.
Now, Ken Barnaby's state of mind is fragile. In a short space of time he has both lost his (only?) daughter and killed a man. Clearly he isn't thinking straight.
Now, Ken knows that he killed the troll, but the police so far only suspect. So what can he do to save himself? The only thing that he thinks links him to the murder is this fingerprint.
In reality, the print might be from his left hand rather than his right. The print might not even be his, but some other random friend that happened to visit the day before.
But because of his fragile state of mind, he reasons that the print is his, and the probability is that it's from his right hand. So he (almost) gets rid of the only evidence linking him to the murder, in his own mind.
Besides, I'm guessing he wouldn't want to lose both hands on the off chance it's a fingerprint from his left hand. I'm sure he'd rather have a hand left (pardon the pun).