Well, being that I am at work and completely bored, I started doing some searches online and I sure can't find much at all. I did find one website that just said to avoid exercise all together. And I found another that suggested that you could have a reaction to antihistimines during a breakout and you should have an epipen with you just in case. I did find a name for your condition though: cholinergic urticaria
There are some interesting articles on the web if you google it, but I sure couldn't find anything about how to treat it other than take Zyrtek or an antihistimine or just don't ever get hot.
This sound like a stupid idea but what if you kept a thin pair of gloves damp and in the freezer you could put them on while working out. It probably wouldn't last for a really long time, but it's a thought. They may have something at a sporting goods store that you could hold on your wrist or in your hand... i've seen heated once for when you're cold but not the other way though so i'm not sure.
good luck to you!