We have a similar problem with rodents and feed grain. Our solution is cats. The white and orange male cats seem to do best and there is some predation, but frankly outside cats seem prone to living fast and furious with relatively short life spans.
Discussion
Cats are definitely on the list but first we need to build some infrastructure for them to hide in. At least a shed for them to be able to kind of get away from predators, give them some what fighting chance at survival.
I'll definitely take note of the orange and white though. My in-laws had a big old black and white male that lived forever and a small tabby female that did as well. But they only had a few predators not as many as we do.