Sorry about that, I tried replying to this post a couple of times but it never went through, so I gave up.
Anyway, I did try it, and it works fine. I left i running for about 40 minutes just to make sure there were no thermal issues, I didn't notice any. When removing the batteries, I thought I smelled something funny, and was afraid that I might have burned something. So I took apart the headlamp, and carefully inspected the circuit to make sure there was no damage. None noticed. Ran it on high for about 5 minutes and check the circuit for any heat. None noticed. Felt the top of the LED for excessive heat (not very scientific, but I was in a hurry), and the LED seemed ok. I think I was just being overly paranoid and imagining it, because after carefully smelling it again I didn't smell anything abnormal. Most likely it was the new battery smell. I'm just glad nobody walked in on me while I was agressively sniffing my headlamp, they would've thought it was pretty weird.
Anyway, I'd say it's pretty safe to run the 712L on rechargable 3.6V batteries. If i have time, I'll try a quick and dirty runtime test during my next night hike. I'm curious to see what happens when the batteries are nearly drained, and whether I'll still have enough power for medium and low modes. In terms of runtimes, I'd probably expect about 2.5 hours on high. The stated nuwai runtimes are pretty accurate for primarly lithiums (5 hours high), so a rechargable has about half the capacity, so about half the runtime.