Which STM32 microcontroller has the most PWM channels?
You can use STM32CubeMX (free application from ST) to search for STM's with a specific amount of timers (I think also including what type, basic, general, advanced etc).

So first find out what kind of timers you need, fill it in the parameterized search within STM32CubeMX and you get a list of all applicable STMs.

Also, for just a few kHz, it might be feasible to not use timers, but do it directly with the CPU (although it will be CPU time costly).