It's distressing when a fox takes a lamb, but it would have been a pretty poorly lamb - or newly born with a distracted ewe - for a fox to have it, so I think of it as it's just nature cleaning up.
Badgers are far more of an issue. They can take down a half-grown Swaledale. You know it's badgers when you find the skin inside out.
So yes, let the foxes coexist with your non-lambing flock, and they'll keep the rabbits at a sensible level. It will actually help the sheep - they will get more grass!
One of my old country-wise 3rd generation hill farmer pals used to feed the foxes that lived around his farm at lambing time, to make sure that no parent fox had to go to extraordinary lengths to feed their cubs. He also supported his local hunt, because he said they only caught the aging or infirm foxes, and those were the ones which caused human sheep- and bird-keepers the majority of their problems. Fit healthy foxes didn't need to risk annoying the humans with their guns and dogs.