I personally feel that bed time is the best time to sweat it out. My thyroid would do it naturally inducing fever. Now, I help it out by sleeping with an electric blanket at low temperature.

Also, if walking makes you sweat, wear twice as much as necessary and make is so you sweat badly.

I also do a bike ride or go to the spa. But I am Pitta, yet I can cope with the hot bath. Strange the reaction you had. Did you put Sea Salt in your hot bath to ease the sweat out?
You my just have a mineral defiency that preeevents you from sweating!!!
Jhan