If you're still wanting to make this work somehow, here are some options I would investigate:
1) Is there a lithium ion button cell that has the same size dimensions as the cells you're looking to use? If so, try two of them together (~7.2 V) with a capacitor in place of the third to take any voltage spikes, and then direct drive an 8x. I doubt the CURRENT could be high enough to hit the diode's max in the first place, you might not fry your diode. Test first though, and make not of the fact that we (myself included) don't generally recommend direct-driving laser diodes.
2) Lighthound sells 10180 lithium ions for $5 plus shipping. See if one of those fits where three button cells normally would in your host. I think it might. In fact I think you might even be able to fit a 10250 (also sold at Lighthound). You'll only get 3.6V, so you'll still need a boost driver, but my sense is that the cell itself could provide more than 210mA, and perhaps enough more, that even after your driver converts some current into voltage, you'll still get your 210mA output.
3) Take a look at combinations of very tiny Lipo cells - even smaller than the cells you get for RC helicopters. They have some that are like 5x10x10 mm. They're low capacity (probably 30mAh for a small enough size to fit two into your host), but they can often do 4 or 6C, meaning 120 to 180mA output. You could try direct driving with two in series, with a capacitor in parallel for protection. Don't underestimate this route. I've tried direct driving a 445 with a 150mAh Lipo thinking there was no way that a 150mAh cell could provide the 2A output required to kill a normal 445. I was wrong.
4) A general hint, there appears to be an underground community of "E-Cigarette Enthusiasts". I don't really understand this - and think it's a bit gross - but for whatever reason, their electronic cigarette gadgets seem to need really high-drain batteries. I've often found myself researching battery technology and stumbling across their forums and tutorials. It's not a bad place to look, because unhealthiness of their hobby aside, I think the general technical requirements of an electronic cigarette (power source, form factor) often parallel those of high powered portable lasers.
I think creative builds, and creative driver setups are phenomenal - and it's great to see members (especially new members!) taking them on
I re-read my earlier reply about the first law of thermodynamics, and realized that it came off a bit snobby, or at least unhelpful. I didn't mean for it to come off that way. I hope some of this helps, and I hope you find a cool solution for what sounds like a really creative build