Is it possible to charge/discharge this cap fast enough?
The main idea of this snubber is to turn off both IGBTs when voltage across capacitor is aprox 1V in order to reduce losses while switching, however, if I add that condition in both cases (with DC Voltage Sources and 12-pulse rectifier) the output current from snubber (red) takes a long time to discharge when the snubber have to track the lower sections of this triangular reference.


So this is your question?
When the transistor is off, current through the contraption should continue to flow, only against an inverse voltage. I.e.: the diodes should conduct and the capacitors discharge; about as fast as they have charged.
So I think there's a fault in your model and/or steering logic.
I don't have the time to analyse the logic and simulation model, so maybe someone else can help you with that.