I sympathise with you OP - when sleep is precious this sort of thing is very difficult to reconcile with the love of animals and birds you do have.....
Anyway, slight de-rail, but Falconry Pest Control is indeed a thing - it's all a bit "nature red in tooth and claw" but it is effective and our local council used them in the Town Centre.
I found this out a few years ago.....
I used to live in an oddly secluded place near the town centre with a big back garden set way back from the road. Our house was behind an ex hotel and there was alot of parking as it had become an HMO.
I woke up one Sunday morning and flung open the curtains to see about 8 various birds of prey, all sitting on their little hoops pushed into our wild lawn, being given their breakfast.
I actually closed and opened the curtains again because I absolutely believed I was hallucinating / dreaming.
I woke my DP and said you're never going to believe this.....
Ten minutes later we're down in the garden, in our dressing gowns meeting owls, hawks, kestrels..... it was utterly magical....
Turns out my adult children had been out in town and met the falconer in a bar. He'd been working but there had been some sort of fubar with his accommodation and there hadn't been a safe place for him to be with the birds overnight - he needed secure parking and to be able to sleep in the van too.
So my kids being my kids, said come back to ours - we have hidden parking and a big garden. We didn't get the heads up because it was the early hours but we really didn't mind at all....
DP died last year, and I treasure the pictures I have of him, nose to nose with a falcon, in his dressing gown and New Rocks..... so I know this is irrelevant to your dilemma but thank you for starting a thread that led me to such a lovely memory.....
And I hope you find the very best solution to your problem for all concerned ☺️