There's no easy solution, unfortunately. I have seen places like this, and wouldn't want my horses living there. However, you obviously don't have the option of going somewhere else since you lease.
I'm not sure the product you posted would be effective, but it's worth a try I guess... see if you can find reviews on them. I feel like they wouldn't stay on very well (I snowshoe and the straps on these would not keep snowshoes on my feet), but maybe I'm wrong. Also be cautious of horses stepping on the plastic edge. You'll want to stay very far away from those hooves -- off to the side and a bit in front. It may even be necessary to ask them to have the horse you lease brought in for you. I know you want to do everything yourself, but this strikes me as a dangerous situation. Getting stuck and falling over in an area full of horses seems risky.