Do you have a chain lead? When I first got my stallion, he had not been handled in about a year other than feet, vaccines. Etc. He was pushy like your mare. I put a chain lead over his nose at first and when he would try to rear or trot off or strike at me, I would pop him in the nose. Not hard enough to hurt, but a quick tighten to let him know that I was not going to accept him misbehaving. As he got better, I moved the chain from over his nose to under it, and now I don’t use it unless I am practicing halter.
One other thing that helped his pushiness was every time he would try to rush off, I would make him back up a few steps really fast. This also worked with my old 16+ hand quarter horse.
One nice thing about the chain is that it goes to the side of the halter rather than under it, so that may help with your reaching problem.
Just a question before I depart, do you have access to a round pen. And does your mare lunge? I found that lunging helps a lot with respect issues, which is what this is.