Lonely is so cute! My four mousers take on rats too, and I feed them and provide water, cat beds which they don't use, and cat toys for them to ignore. Mine seem to eat pretty much all of the rodents except for the front teeth of the larger rats, for some reason. I'll find a blood splotch on the concrete with a set of rodent teeth and know they got one.
One of the joys of buying a former hoarder house out in the country.
Sorry to hear about the pigeon.
I think the reason why they leave the rat teeth is because rat's have serrated edges on their teeth.