Can you put meat in your green bin? We have a brown one for vegetable/garden waste but you can’t put meat in it.
Other than that, and sorry if this is a bit obvious, have you tried putting a bell on the cat’s collar? You can get them from pet shops and it stopped my friend’s cat sneaking up on so many birds and baby rabbits.
Or if it still catches stuff, just put them in a bin bag in your normal wheelie bin, bin ben don’t know what’s in there then.