Just a thought, I wonder if wrestling shoes would do good. The sole is very grippy on slick surfaces and there is very little if any padding on the sole that will deform under load.
Next I might suggest a crossfit shoe. I have a pair I used to workout in. They are heavier duty and more supportive to the top of your foot than a running shoe and grippier. Made without excess padding which effects stability. Also great for squatting and deadlifts.
I have not tried my wrestling shoes during a crossfit wad but would love to. I just don't do crossfit anymore as it get hurt too much. When I do burpees I normally have a pool of sweat under me. The corssfit shoes still grip while running shoes slip sometimes on the rubberized floor of my gym or the canvas of a boxing ring.
We did tire flips and drags with a 500+ lbs tire and my old crossfit shoes were good but we were on concrete, zero slippage. I had an old version of the reebok nano, they were way cheaper than the nike or crossfit brands.