Those small 3255 amps are simply not able to run high power for more than a few moments because of construction. The Chinese love to build cheap things that sell well and amaze on first sigth. Terminal function is secondary. So these are about 1/3 the size such an amp chip would need with passive cooling, saving on material, wheight and shipping.
Anyway, many HIFI amps are build just the same: High output, too little cooling capacity. AVR's have build in fans and usually external, active subwoofer, so the main load is bass free under 80 Hz .
My advice for you is to get a low cost PA amp for the bass as your speaker does bi-amping. Use the Fosi or the like for the mid-high input. That way you keep the sound quality and add permanent power where it counts.
The mentioned Behringer may not be the right thing, it is a "studio amp" with passive cooling and may not permanently deliver what you need. Get some amp with a switched or regulated fan, even if it is loud, you won't hear it when it kicks in at deafening SP levels.
If the 3255 still runs too hot, get a second Fosi one and run it in Mono mode. That should help.