High power tolerant (10-20 W) LC filter
TL/DR: use a big film cap, like a motor cap.


  DC also creates magnetic background that I want to avoid, so that method is ruled out.


Use a non-inductive pattern for your nichrome wire, this will cancel the magnetic field out.


  What do you think went wrong with my attempt? e.g. Was a 35 V capacitor rating not high enough, or was cooling needed?


A 100µF general purpose cap has about 1-2 ohms ESR, which is resistance so it will convert current into heat. This creates enough heat to explode the capacitor. It is quite small, it can't dissipate too much heat.

You need a low-ESR cap. A good idea would be a cap specified for a ripple current higher than the AC current you run through it. (that's the idea with the ripple current spec). 

Ceramic has voltage-dependent capacitance, but the voltage across the cap will be small. So, ceramic may be an option. You can use any parallel combination of caps optimized for lower cost, like 10x 10µF 25V caps. Be aware that it will add a bit of distortion on the waveform.

It will be tricky to find 100µF aluminium caps with low enough ESR and high enough ripple current, but you can use polymer caps, which should work well.