Grilling and eating it — that's certainly the best way to dispose of a pigeon —
IF IT IS NOT A FERAL PIGEON —
raised on cr@8 picked up from the street,
riddled with deseases and parasites.

Those don't go on the grill....
A non-transparent double plastic bag is often a good start. It goes where the other plastic bags go.... At least where I live.
But you're in Chicago — I don't know how much they are into recycling and stuff.... Better find that out first.... 🤣

Matthias