as far as i am aware,if you are removing vermin, with out charge/ voluntary,your shooting insurance will cover you.
the allotment owner rents the land from the council,making them the tenant.which i think is all you need, is permission from the tenant,to shoot quarry under the license.
However!
you have stated pigeon shooting! are they feral or wood pigeon?
feral are a general pest,and can be shot in day light/night time.wood pigeon,is a different matter,it has to be protecting crops/vegetable's i believe?
and deffo no shooting in your garden,unless you have a vegetable