"Timing belts are just not designed for such high speeds" This statement is inaccurate. For those who prefer observation over engineering specs, note the black thing on the front a blown top fuel dragster engine . . . 10,000 RPM 6" dia pulley . . do the math . . . or the primary drive of a custom Chopper (motorcycle), or the dry sump oil pump on indy cars or perhaps just the cam drive on many modern automobile engines. In machine tools, some surface grinders have 'timing belt' drives on the grinding wheels.
And . . . all of these real world examples use the OLD trapezoid tooth profile. The newer curvilinear profiles are far better. Flat belts have been used for years for high speed applications like tool post grinders and so on because they are less prone to whipping like V belts which have a much larger tangential cross section. A 'timing belt' is similar to a flat belt in dynamics, except that there is no slippage, which coincidentally is actually where the heat comes from in v-belts, round belts and flat belts.
Round belts come in two types; elastic and corded. Example of elastic are vacuum cleaner belts. Belts with chords don't stretch (much) and you must follow the manuf specs for the type of belt . . specifically the type of chord.
In any case, no matter what belt drive setup is chosen, the amount of heat contributed by the belt is minuscule. The heat comes from the bearings.