PMD, that's definitely a tough situation. Chasing it could cause it to hurt itself thus, more suffering. And i know here or more specifically, in the city of Seattle, shooting a Seagull injured or not is something like a $5000 fine and up to a year in jail (I could be wrong about that but you get the idea). It could require a little more stealth on the human side since no one seems to be all that anxious to assist in the situation. A pump action Pellet pistol perhaps? If you could get within 20 feet of the bird, that could end it. Hopefully the bird humane people will come and all will end well.
Here's the flip side... If it were a Bald Eagle, you would have every known (and even some unknown) gubment agency their to save the bird, believe me, I know. We had a Bald Eagle nest on our property many years ago, and I was cleaning debris when i saw a head and beak look up at me. The baby eagle was flapping its wings jumping up and down in the nest when the nest fell apart and the youngster fell down. The parent eagles were distressed flying around in circles. I didn't know what to do, of course it was a Sunday, but I found the number to a Wildlife rescue organization, told her what happened and within 3 hours there were 5 vehicles from three different organizations and about 15 people in our driveway.
Well, to make a very long (and very cool) story short, the next day this tree/forest/wildlife expert came out and rebuilt the nest using rope and lumber I had laying around. They put Schuppa (we named the baby) in a rook sack pulled him up with a rope and put the adolescent eagle back in the nest. First time ever in the state of Washington. Once he climbed down, he climbed up another tree about 50 feet away and mounted a remote control camera and handed me the button. They wanted pictures of the parents feeding the youngster to ensure it wasn't rejected. Schuppa was not rejected, and as far as I know he is still doing well somewhere over there on Puget Sound.
Sorry for the tangent, but it is just a success story that actually happened and we were lucky enough to be apart of.