Yep, it definitely sounds brake related. I've seen it before, more than once. Once the brakes get that hot, they need to be replaced, as you cook the binders out of the pad material, and make a mess of the rotors and caliper seals. If you must, change just pads and hoses, but check, clean, and lubricate the calipers really carefully, and keep an eye on them. Then dont be surprised if the calipers either seize or leak in the near future, or if the rotors start to flake off a layer about 1/16th of an inch thick. Unless you are able to keep an eye on it, I'd seriously consider doing it all, rotors, pads, calipers, hoses, and possibly even master. FWIW, I have had multiple rebuilt masters fail in the first couple months. I will not use a rebuilt master on my own, and I pretty much refuse to install them into customers cars, too. The ceramic pads (most of the premium stuff nowadays) work well, and dont seem to have as many issues, but I've seen way too many comebacks for noise complaints with them, even properly installed with a nice layer of anti-squeal. If you can talk to the parts guys, I'd try to get a decent semi-metallic, and put up with a little more dust, and possibly more fade if they are really hot, given what I'm guessing about the owner.
As an aside, I had an f150 that I had the brakes glowing and smoking something fierce (might have been flames, I was afraid to look, they were lighting up the sides of the road!), towing a boat with no trailer brakes through a notorious peice of mountain road (duffey lake, for those who know it). The only thing that put them out was the puddles at the bottom of the hill, which shatterred the pad material and cracked/checked the rotors. I rolled the last 100km on rivets into the cracked rotors, real slow, but at 1:30 am in Mt. Currie I just wanted to get home.