As TorchBoy clearly presented, a buck driver generally will have too high of a voltage overhead to run in full regulation on 1xLi-ion. There are a few low voltage overhead buck drivers that might work -- they all have an overhead of about 0.5V. With an XP-E, you might get lucky and get a low Vf of around 3.2V at 1A drive (note, the XP-E is actually spec'ed at a max of 700mA; if this is an issue, you might consider an XP-G instead). Thus, a relatively large sized Li-ion like an 18650 probably can hold its voltage under load long enough to get some useful run time in regulation. But a smaller cell like a 16340 might not. And if you get a more typical Vf of say 3.3V to 3.4V at 1A, then you start to cut into your run time in regulation even when using big Li-ions.
As a concrete example, I can tell you that the buck driver in choice #5 in post 2 uses a buck IC with a recommended min input voltage of 5V. That isn't going to work very well with 1xLi-ion. I've tested several other buck drivers (different from choice #5) also based on the same buck IC and I've typically gotten a voltage overhead of about 1V to reach regulation. That's too much for 1xLi-ion.
For any of the boost drivers that were suggested in post 2, they won't work well either with 1xLi-ion. The driver won't be in boost mode, except at the end of the Li-ion's run time life when its voltage under load falls below the LED's Vf. But you probably won't want to drive the Li-ion to such a low voltage anyway, which is bad for cycle life. Instead, you will be direct-driving the XP-E for probably 100% of the battery's operational life. And with the low Vf of that LED, you will be overdriving it by a fair amount. See
this post for example. If you have an XP-E with Vf of about 3.4V at 1A drive current, then pushing the XP-E with 3.7V (the nominal voltage for a Li-ion) means over 2A drive current. The XP-E will be very unhappy with that unless you have really good thermal management.
I've used many AMC7135-based drivers in 1xLi-ion applications and they work very nicely -- fully regulated output from 1xLi-ion with long and efficient run time. I've used these drivers to run Cree XR-Es, XP-Es, and XP-Gs. Most of these were single LED setups but some of them include triple XP-Es and XP-Gs.