I walked the Two Moors Way with Jodie, a GSD (oh yes, and one of my mates as well). Her problem was sheep which wasn't a problem until the middle of Exmoor, but we got away with that one. We also lost her for a while when we rounded a corner on a lane in 'middle' Devon and found ourselves face to face with a stag. She wasn't vicious though and never hurt anything, my concern was always that the animal might be bothered.
There aren't many cows on Dartmoor and Exmoor, but I think what I'm getting at is that if you take a dog on a LDP, there is always a risk that it will chase something. You just need to mitigate the risk.
But the other side of the coin is that a dog can be a wonderful walking partner.
BTW, as with your situation Jodie wasn't my dog either. She belonged to a very good friend who understood that when she went walking with me, for her it was like two weeks in Barbados for him. Despite the occasional scare, I wouldn't have missed it though.