Honestly, though, they don't allow you to hang a hay bag for your own horse??? What the heck ???
There is a huge variety of very sturdy small hole hay nets that have been used safely for many years and thousands of horses. With to many benefits to count. Just make sure it hangs so they can't tangle in nothing or paw at it. Then its no "un-safer" then the very fence she is behind and snacking on.
To me, this would be non negotiable.
My mare is in the very setup you describe. She has a large paddock with a large attached stall (open door always). I hang 3 slow down hay nets. Well, 2 are fastened in huge rubber totes so she eats with he head down and one I hang. I have several areas I can move those totes and bags too (made hooks) depending on the weather, inside, outside, shade, sun, etc. She moves nonstop from one bag to another, "grazing" along. It makes a HUGE difference to their physical and mental health.
Hay bags will NOT damage their property and it is YOUR horse. Its non of their business to decide what is dangerous to your horse, unless it is dangerous to their property or themselves. Your horse would benefit greatly from a couple of bags to move in between and it would be all the enrichment she needs. Nothing else can take its place, really. There are some "treat balls" that dispense treats when rolled but honestly, I have yet to find a mare (unless young) that "wastes" much time at playing, LOL.
Have another talk with the barn owners, tell them it is affecting your horse negatively not to have something to chew on between feedings and you want to provide slow down haynets. I find it hard to believe they would rather throw you out. And if they do, well, that is not a place I would stay at anyway.