My thoughts (but not up with the American regs):
Track to the missed approach point on climb and conduct missed approach as one of your navaids has failed (DME) and you are now below your published MDA. Go-around and conduct another approach using the higher minimas.
The navaid has failed.
What happens if you become visual at this lower altitude? If you maintain all the requirements for a visual circling approach (ie. constant sight of ground/water, clear of cloud, etc) AND you maintain your obstacle clearance - go for it!
This is all in a perfect world. In the real world .......