And, consider that you work in a cat shelter. What happens the next time you feel sympathy for a really sweet cat? Maybe it's time to consider limits?
I understand this, but I've been volunteering at this cat shelter for almost two years. I've never felt attached to a cat like Jack. Not even the cat who comes when you call his name or the cat who willingly jumps on your lap. Sorry if I sound a little rude, I just... well, you know, want Jack...

You already have 3 established cats? You are assuming that Frank would be the one to take offense. Maybe any one of your cats (everybody's been neutered for years, right?) might take offense. Or maybe the two females who are buddies, might gang up on the new cat.
This made me think and I don't think the females would gang up on Jack. The older one, Polly, just lies in her basket most of the day and never fought with The younger one, Taffy, when she first came. It was either Taffy who started the "fights" (she probably thought she was just playing) or Frank. Frank usually starts the play fights with Taffy. But Frank has been know to come home form outside with a few scratches... I don't know if it is from other cats or from just wiggling into tight places.

And yes they have all been neutered for years.