I've done Red Pike to Haystacks with a dog, and Fleetwith to Haystacks with a different dog. There were no special problems. I also once did the whole Buttermere round with a dog too with no problems. Depends on the dog, obviously; mine were both collie crosses who were pretty good on hills and able to cover a lot more miles and contours than I could.. If you go Red Pike - Haystacks, there's a steep descent off High Crag which at the time was quite loose and the dog needed to be off the lead. This might have been "fixed" by Fix The Fells (although some of their fixes are more awkward than they were before they were fixed) (slaps own legs for blasphemy). I've not been along there recently. But the point is that the dog needs to be able to ignore sheep because it might be awkward on the lead on that particular section.
Haystacks to Fleetwith is very straightforward but the descent off Fleetwith to Gatesgarth, if you're going that way, is also very steep with a crag to avoid at the bottom (has a white cross on it where a Victorian lady fell off)