I don't know how your barn is constructed but I am assuming that it will be 2x4 framed under the floor, spaced at either 1.5 ft or 2 ft on centre, with the plywood laid on top of the frame. If that is the case I would suggest two things. One will be more doable than the other for you but the end result will be the same - to ensure your little charge does not step through the plywood. 1) Add additional 2x4s to the floor frame under the plywood. I would suggest a quantity to recreate spacing a couple of inches apart. The reason for this is that the original spacing will allow the plywood to flex or bend with concentrated weight (as well as plywood does have a limited weight bearing capability) and then break. 2) Place and nail/screw 2x4s (or 2x6s - whatever is readily available) on top of the plywood just like you would lay a hardwood floor. The 2x4s are snugged up against each other and the 4" side touches the plywood (not the 2" side). Also put them opposite to the framed pieces - eg if the framed 2x4s run north/south, then the top ones run east/west. In either of these cases, use the sealant on the plywood and the 2x4s if you can't get pressure treated ones.
For mats, I'm assuming your in the States, so you could go to Tractor Supply and get them there. I believe I read in an earlier post (dealing with horse trailers) that they have matting on roles, which is a 1/2" or less in thickness, that is cut to your specs. I'm sure that would do quite well for you. The mats I use are quite thick and extremely heavy but my horses also have another six or seven hundred pounds on your fellow.
With regard to the concrete, I personally wouldn't go that route. I think if you do the sealant treatment (which isn't hard - it's just painted on) and the mats you'll be fine.