My sister and her husband moved into a place that had an old style barn (loft in it which they don't use for fire safety reasons) a number of years ago. The walls are about 8 ft high and the floor is concrete with stalls, at the time, set up for an assortment of farm animals. At one point, they were considering jackhammering and removing the concrete from the stall areas, filling with dirt but leave the concrete in the alley. In the end, however, after further debating and pondering, they left the concrete in place and put rubber mats over the concrete with bedding on top of that. I know one of her horses is a 16 hander but wall height hasn't been an issue to date with this set up. So while your situation may not be your ideal, if you stay away from situations where a horse may rear, you may do alright.
Good luck with the conversion (and dare we ask for photos?).