Perhaps you could find out if there is a local pigeon fanciers club Greyduster and ask if anyone would be willing to come to your garden to try to catch it.
We had a homing pigeon resting on our back bedroom windowsill for a few days 2 or 3 years ago. It eventually flew off, but only after I gradually and gently opened the window out, inch by inch, thereby reducing the amount of room on the sill for him to perch! I think he was just resting and, once recovered, was happy to continue his journey. Or her journey, mustn't be sexist 
I am procrastinating and need to stop!
Welsh Senedd Election - PR in action. This will be interesting!
