What methods can be used to remove ringing artifacts in the output of a digital filter?
Sticking with linear systems, removing the ringing is nearly the same as adding back some of the spectral content that your really steep transition filters removed.  Why use some crazy scheme to add back the stuff in the "softer" transitions that your hard-edged filters cut out?  Just use a more reasonable total filter response in the first place.

Going to non-linear systems, you could use some sort of AI pattern matching to determine what kind of sound waveforms would be perceived by a human a ringing, and just gate those waveforms out.  But that might just add weird sounding artifacts of its own (as well as also spreading spectral content outside of your really steep filters).