IMHO, if you walked up Haystacks via Wharnscale Bottom (the path running south east beneath Fleetwith Pike) then that would present no problem to your four-legged friend.
The route up Haystacks from Scarth Gap could be a little trickier for Rover, given that there are a few small 'scrambly' bits. I think a dog could make it but I've not taken one up myself