You're lowest heat, yet still awesome option is a nice simple SET. You don't need muc power but any class A SS amp will have more watts than you need and all the heat associated with them.
Your BAT uses the 6C33C tube that is one of the hottest ones out there. It runs about twice the temperature and radiated heat goes as Texp4 so the increased tempeatrure adds a lot of radiated heat. Also the BAT makes much more power than you need. A imple 300B or 2A3 amp will be less heat than a TV screen, perhaps 100 watts (heat, not sound). Your BAT is likely 3-400 watts.
An even lower power option is the Decware SE84series which is about 50 watts and plenty of power to drive those speakers.
Jerry