The best idea I have seen was a large bag made from fine mesh curtain material.
It had a drawstring at the neck, and you simply popped the pigeon in, and pulled the string tight.
Flies could still lay their eggs on the outside, but at the end of the day, you just took the birds out and stored them in the normal way.
The other trick is take a length of hessian sacking soaked in cold water, under which you put the shot birds.
This can work for short periods, but flies will eventually find their way in.