Boots are definitely the best compromise for now. A dirt arena is fine as long as it is well maintained. My home arena is basically a flat spot with a fence around it, no extra footing, but we run the tractor disk over it 2x a year to really scratch it up and then run a 4-wheeler with a chain drag over it at least once a week. Works for us.
That's really abysmal sounding arena maintenance. If it isn't just a logistical issue because of the season, I would start looking into a different place to board. Over time intensive schooling over ground that can trip a horse up can be rough on their legs, on top of hindering her in learning how to use herself properly. She can't stride out and move if she's constantly watching her feet to keep from tripping.
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown