Naturally, these conditions exist right where you feed the birds - so don't feed the birds. The first order of the day is to control the food and water supply. Eliminate the food supply, clean up the place so they will have no source of food. If you can eliminate the water supply, do that too. Replace their food supply with bait.
Rodent bait takes a period of time to work.
So the bait is designed to work slowly, so the rats do not connect the bait with its effects. If you can also control the water supply, there are liquid baits that can be used. Liquid baits, however, are much more likely to be consumed by non target animals, so great care must be taken when using liquid baits. Food baits won't work if the rats can get a regular food supply. Oh, you may get a few, but you won't get all of them.