Any question answered...

It's legal to do it by conventional means (e.g. shotgun*, peregrine falcon, cat), but anything too unconventional might get you into trouble on animal cruelty grounds.

*Assuming you have a licence for the shotgun, and operate it safely. I don't know the exact laws and regulations on destroying vermin on your own land with fire arms, but I'm guessing any downrange projectiles landing on other people's property is frowned upon :-)

I was thinking more some kind of pigeon I.E.D placed in the gutter above my window where it likes to position itself for maximum annoyance.

Or the Roald Dahl drugged raisins might go down well...

Little fucker. It's an aggy wee bastard too-tries to beat all the other pigeons up. Begging for a slap down it is.