Breathable painting for acoustic panels?
I think your best choice would be to have the image painted first (on primed canvas, or whatever it takes), scan it/take a high quiality photograph of it, and have it printed onto the panel. 

Notes:


I have personally never worked with acoustic panels.
Yes, there are breathable materials you can paint/draw with (like ink, watercolours).
I couldn't find what material is used for acoustic panels.
There is a possibility of the painting materials affecting the quality of the sound absorption of the panel regardless of them being breathable. 
Apparently professionals print onto the panels, so it is know that this won't affect the quality of the panel.
Having the image painted first by all the necessary means will also, most likely, give you a better result imagewise.