Hellosailor, thanks for the particulars.
I know one thing. In my new job (Airfield Management) i'm in charge of the BASH (Bird/Wildlife Aircraft Strike Hazard) program to aggresively keep Birds/Wildlife off the airfield for aircraft safety.
I have bio-accoustics, screamers, bangers and a big truck. It takes everything i can to keep them out. I use both passive and active controls in my program. (You would be surprised what you learn about passive wildlife control)
The #1 thing you are told is to never let them become established. They will tell there buddies and be back. #2 THE most effective method of control is a shotgun. As you said, the US wildlife service dictates when and how this can be administered. Never happens. We use this technique overseas for the most part. I have a very expensive bio-acoustics machine, It sucks. Save your money.
Most birds are a lot smarter than people give them credit. We have found the best active controls besides chasing (truck) screamers and bangers is to use a dog that actively chases. We paid $15k a summer to have a border collie come out. Next best, Falcons. Next, remote control helicopters.
I like the cat idea the best. You could keep it on a invisible fence.
Honestly, Look around. Find out why they like the local area so much. Where do they eat? Get rid of it. Where do they nest? do away with it.
Birds don't like what they can't see, they are looking for there predators. We let the grass grow tall. (7-14 inches)Plant thick shrubs along any water edges. Manage trees to discourage roosting and nesting. Why are they loafing at your location? Are they in between feedings?