Well, to the OP:
You can run a Biopace ring with no trouble. All you need to do is find the tightest point of the rotation (as you seem to have done ), and then set the chain tension to be a little (quarter inch or so) slack at that point. You should be able to run it without any trouble.

I have run a Biopace (46/16 ratio) for almost a year on one of my fixed-gear bikes. It is quite a nice ride. I've never had any trouble, and the chain has stayed on. I sometimes go to quite high cadences (over 120 rpm; sometimes way over, depending on the hill), with no problems. The pedal-stroke is very smooth, though I cannot claim any performance advantage over a round ring. I do like the Biopace better, however.

Trackstands become more problematic, but are still possible. I use brakes, so I don't know about how my setup would react to skidding. On the workstand it looks really strange as the chain angle alternates, but I don't care much.

Aesthetics aside, there is no reason why you couldn't run Biopace on your fixed-gear.