I doubt this would work with a 355nm laser, 355 is just at the edge of transmission for normal glass and goes right through good quartz. You would need to go to sub-300nm (266nm would be your best bet) to be able to do any decent UV micromachining. Alternatively, a CO2 laser might do the job, ideally one of the big pulsed ones but I think even a normal ~100w should have enough power to vaporize glass if you focus tightly enough.
Another alternative would be to sand blast it, I am not sure if you will get the repeatability/resolution that you need, but if you use a laser cut stencil (ex, sold for PCB solder past application) and start blasting away it seems possible that you could get what you need.
On an unrelated note, is there a particular reason that you want to do the shaping on a finished mirror? It seems like it would be much easier to do it on a blank, polish, and have it aluminized just like you would with a telescope mirror.

