I agree that it sounds as if the mare is scared, and not of the person standing by the arena. She’s probably got pain issues and because of them, she’s probably been ill treated. As much as you like her, you can’t fix her unless you buy her and that doesn’t sound like a good idea.
I’d be exploring other options. For whatever reason, they simply don’t have anything available for you to lease or that you even really enjoy doing lessons on. Their over eager attitude in trying to sell you on the lease of the mare makes me very uncomfortable. It makes me question how suitable a match they might find for you.
Check out some other barns. You don’t have to tell them where you currently ride but do be honest about your abilities and fears and goals. Try to arrange to be there at a time when there are lots of lessons, like a Saturday morning. That way you can see the schoolies and the instructors doing their jobs. Ask if they do part leases on schoolies and which ones might be available. Remember, not having suitable horse for you doesn’t make them a bad barn, it makes them wrong for you at that moment. At places which lease month to month, things change quickly. The high schooler who is currently leasing the nicest schoolie in the string might be graduating in May and heading away to college, leaving the horse available. Give a barn a chance if you like it but the “right” horse isn’t available for lease.