All the usual ooptions
Shoot them, trap them, scare them, block their perches and nesting areas, release raptors to snack on them, use drones to harass them - all of these have been done with varying degrees of success for similar problems elsewhere. What is called for here is something a bit different... something unique. To that end, please vote on the following or suggest your own solution that may in some way have a connection to the issue (or not).
1) Microwave blasts to fry the flying beasts while in the air. Not as far-fetched as you might think. Radar will accomplish this if used (in)correctly.
2) Declare war on the bastards! I know declaring war on things is more of an American thing, but it has worked fairly well for us. Stage a WWII re-enactment themed fumigation the entire area. There must be some vintage aircraft that could be used to drop gas canisters onto the benighted area.
3) Open the town as a cat sanctuary while running a simultaneous campaign to encourage cat ladies from all over the world to bring their pussies to have a good time. If only one or two decide to do this, the problem of roaming cats will quickly displace that of dive-bombing pigeons. Dogs next, followed by goats, cows and horses.