Hello.
(Sorry, but would you mind writing in paragraphs? Would be really easier to read you! Thank you!!)
Look... I am really, really sorry you are dealing with that...
Nevertheless : we have to aknowledge living in a specific area, such as rural area, don't give people the right to ignore basic hygiene.
Indeed : if you settle somewhere, you need to respect the people who were already living here before you...
(I am NOT telling you don't respect them, of course...!! But if ever complaining people are in good faith, you could want to put yourself in their shoes - just to understand they may not be pain in the neck without reason...)
(Sorry : will be a bit long below...)
...I live the countryside. In a rural, agricultural area...
(Not very far from you, since I'm from France...!)
We have rats in the lands, of course... but having rats because we live in the countryside is different than having a RISE in the rat population because some people can do so much nonsense they are attracting the rats in the area.
Indeed : people, especially newcomers, can clearly be the reason quality of life is declining in the countryside...
...Some years ago, rats were literally running inside of my house because of my neighbours : they had the bright idea to adopt chickens... but without being able to (know how to) take care of them correctly!
These people were eventually responsible for a big rat infestation in MY home...!
Of course, that is not because I live in the countryside that I am a dirty person, and so, that I am happy to be infested by rats because of others...!
(Thanksfully, I am not afraid of rats... because otherwise...)
But that's not all... Another example, just to insist :
...In 2018, in another area (since I had moved then) : a new family came to settle next to my house...
These people, it was clear from the first weeks they settled in, lived on traffic of all kinds...
So, obviously, they had been trying to hide themselves from view - and they were so paranoid they did so openly!
Thus : in order to get my family and me, and another neighbour, out of the neighbourhood - so they could keep doing what they doing without being noticed by people -, these people put goats just A FEW METERS from my BEDROOM...
And I have animals, and I love goats; but trust me : when the smell is so bad you can not sleep or even eat because you have smell of diarrhea in your bed... your health can eventually take a hit so strong you want to cry.
Living (then) in the countryside did not give them the right to make my life miserable - even by using farm animals : I am a person, and I was here BEFORE them; and so, I think I deserve(d) some basic respect.
(I can take smell from all kind of animals, but goats near my bedroom was just TOO MUCH...! And my house is surrounded by cows, so trust me : I'm used to strong smells!!)
In addition : because of these goats, I had been so infested by rats (and flies, among others) even all the cats, that killed several rats per day for each one of them, could NOT stop the rats from running in my garden and destroying my vegetable garden anymore!
(I lost so much money because of that...!!)
Eventually, the gendarmes made the new neighbours remove the goats : their installation was illegal, so they did not even have the right to have them there...
But : these people had next the nerve to take ponies to replace their goat! Again : installation being illegal of course!
Obviously, they STILL did not take care of their (poor) animals; and so, the rat population that increased because of them did not decrease - in spite of the fact the goats were not there anymore...
Rat population in my area only began to decrease some months ago : the man was arrested and put in prison, so the woman had to give away the ponies.
She actually even ran away from the village...
...I love animals...
I HAVE animals. But I never, never provoked a rise in the rat population despite the number of animals I own!
...If you are responsible of a rise in rat population, you really need to find a way to deal with it. 
Not only because of your neighbours, but also for your birds' health...!
Where do you put your birds' food?
You could at least want to use wall-mounted feeders, so rats could not get to the food...
(Don't put food on the ground!!)
If rats are NOT attracted to your garden, their population will decrease eventually - especially if you have cats near your house to hunt them down...
And less rats in your garden mean less rats in your neighbours' gardens, and so : everybody will be happy!
...At least, think about inspection : you don't want to have to separate from your birds because of your neighbours if you are not in the wrong, right?
IF you are responsible OR NOT of some rise in the rat population would not really be THE problem with an inspection : indeed, fact is you will have to show that the rats CAN NOT be attracted to your garden in spite of your animals - meaning you really would not be the one responsible of the rat infestation!
So you will be sure to not have any problem aftermath...
Try to talk to your neighbours.
The kindest ones could hide (from you) the fact they have had more rats in their gardens since you are settled next to them... because they would not be so daring to tell you the truth!
Some people are really so kind (OR even so cowardly)...
Good luck!!!
(Also, you could want to know rats, being mammals, apparently hate strong peppers...?
If you can't buy wall-mounted feeders, why not just put some seeds/pellets MIXED with powedered Cayenne peppers on the ground... so rats can eat them AND suffer because of peppers... and so they won't want to come again in your garden to - maybe - steal food from your birds?
If I had a rat infestation AND no other solution for the time being, I would actually try this method at least for two-three days - until I find better feeders...!
But I don't know if it really works...)
NOTE : If you are NOT responsible for a rise in the rat population on your area... really GOOD LUCK to prove it!