One wonders about the role of the landowner of the field in all of this. The simplest thing to do is for you to tell him/her that someone is trying to nick his land by Adverse Possession! Then he reclaims his land back before he loses it - and you get your dog-walk field back.
At least you do until the true owner of the field decides that he/she doesn't want you in their field anymore.
You have no other options, because, from what you say, you are not going to be able to claim a private right of way over the field because you are not using it as a 'path' to get from your gate to a road the other side. You seem to be using the access to merely walk around the field with your dog. You would need to demonstrate use for over 20 years as a 'path' from A to B to stand a chance of acquiring a private right of way.