I'm thinking - because I tried everything '- to make his bills / correspondence disappear
I don’t know how it is where you’re from, but here mail theft is a crime punishable by up to 10 years in jail. If it’s also a crime where you’re from, is this something you really want to risk?
You say you’ve tried “everything”, but only mention talking to the neighbour a few times. Have you talked to the owner of the building? If the pigeons are really making that much of a mess, the owners might not be too pleased about it either, and they’d actually be in a position to do something about it.