'Many vegetarians (vegans) believe that Jesus was a vegetarian, and these verses are used to "prove" that He wasn't, for they indicate that He ate fish.'
Yes, they indicate that Jesus ate fish, but they don't prove it. The fish eating narrative is no a passing comment of little significance, rather, it's a deliberate insertion meant to convey a specific message. It's for the same reason that we have Thomas being told to put his fingers in the wounds of Jesus - both narratives are meant to counteract the gnostic claim that Jesus had risen spiritually and not physically.
Jesus insists 'touch my wounds', and he eats fish, like any physical being, and, most importantly, after He is risen. No, this narrative is not 'prove' of Jesus consuming the body of a dead creature - it is meant to depict a physical and not spiritual resurrection.
According to Matthew and Mark Jesus was not even in Jerusalem on the Sunday after His death, where he allegedly ate fish. All of the resurrection narratives, including the messages of the angles to the women at the tomb of Jesus, was that he had gone before them into Galilee. This is where he told all of his apostles that he would have His first meeting with them after His resurrection. Luke's fish story takes place in Jerusalem and not Galilee; it is counter claim propaganda that Jesus was risen physically in body, evidenced by Thomas and the claim that Jesus ate fish. There is no other scriptural reference that Jesus actually ate the flesh of dead animals. I think it's time that we started admitting this and not doing Jesus the disservice of making him a stakeholder in, and excuse for, animal slaughter and abuse.
Incidentally, in reference to Jesus' assertion that not one sparrow shall fall from the sky that His Father would not know about (paraphrased). Did you know that sparrows were often caught, cooked and eaten as small pieces of meat on sticks. Most carnivorous Christians don't know this; it casts rather a different light on this story.
How wonderful to have a wholly loving and compassionate Jesus who embraces and respects all creatures great and small and doesn't condone animal abuse, exploitation and slaughter - especially in an age when vegetarianism is so easy to follow and flesh consumption is no more that an addiction.
The reason why Christians, even vegetarian Christians, insist that Jesus ate fish is because of the above mentioned reasons. The Christian faith relies on a physical resurrection, and the fish claim, as well as the Thomas story, are insertions in support of this claim. They are propaganda, in this day and age we would call it 'spin.'