I've run it for 30 min straight, driven by 1xAW17670 in the original SF L4 body (old E2 Executive twisty non-lockout tailcap). Initial tail current draw was about 1.2A. This is the same tail current draw I measured in an E1L body using 1xIMR16340.
The KL4 head gets warm after about a minute and is noticeably hot within about 5 min. By 15 min or so, the head is burning hot to the touch but there is no tint shift. In some previous thermocouple tests on a KT1 and KT4, a burning hot flashlight surface measured to around 130F (54C), and the heat sink temp wasn't excessive (162F max for a room temp ambient test -- could be different if you start out at 100F ambient).
At the end of the 30 min run, the hot spot lux measurement was ~96% of the initial value. So thermal management seems sufficient. In contrast, with the stock Luxeon V, the hot spot lux value continuously drifted downward. It dropped about 8% within just a few minutes IIRC. If you look at the photo of the Lux V emitter slug in my original post, I think that the thermal interface to the heat sink was poor at best. The LED was poorly secured to the heat sink. In fact, if you pressed down on the Lux V case, the LED would actually sink perceptibly downward. But no thermal paste got pushed out, so I doubt that there was much paste present to couple the slug to the sink. And even if there had been enough paste present, it would seem that the gap between the slug and heat sink was overly wide and thus the amount of thermal paste would have been overly thick anyway.
To get more current sent to the LED, you could use a different driver. TaskLED's Fatman looks like one candidate. The stock driver is about 0.87" diameter. The Fatman is 0.8" diam. If you want to mod the stock driver, then you'd have to change the sense resistor value to get more current out.
At some undetermined point, more current to the LED might be counterproductive due to excessive heat generation. If you use a 1000mA boost driver, for example, the MC-E is going to draw around 6.6W of power, vs about 4W for 640mA. IMO, that's a lot of extra wattage that is mostly going to turn into waste heat. The small KL4 head is already running hot as-is.
Update: I made a full one hour run using 1xAW17670. The cell started running out of gas very soon after the 60 min mark (the lux values started dropping clearly faster than for the first hour, although not a drastic fall-off), so I terminated the test so as not to overdischarge the cell. Even so, the OCV immediately upon removal from the light was 3.37V.
After a short delay, I put a thermocouple to the side of the light and measured 130F (felt burning hot). The light was very difficult to hold in my hand at the end of the test. Presumably, the heat sink was even hotter and the LED junction temp hotter still. However, I noted no tint shift and the hot spot lux at the 60 min mark was still 95% of the initial value.