Ripping off its jaw is a pretty brutal injury, even if it's unintentional. That'd be something you should euthanize.
That said, I don't think fishing is *as* bad as some other things, just because the same fish may strike a lure/bait multiple times during the same day. I remember going fishing once when I was a kid, and I caught a bluegill that had some damage to one of its pectoral fins, obvious enough that one of the chaperones (it was a school field trip (at Litchfield Lake, if anyone from AZ is familiar with that tiny body of water) noticed it, and came over and commented on it. That same bluegill was caught 4 more times that I heard, including by a kid next to me. They may just not have a lot of sensitivity in the jaw area. That would make sense for predators, because the prey may fight back, and feeling residual pain, or even too much initial pain may lessen the ability to hunt, causing the fish to go hungry, and possibly starve.
Just to be clear, I'm not saying they don't feel pain, just that they *may* have fewer pain receptors in their jaw area, which may prompt some people to believe they don't feel pain.