Note that the 202 lpw from your leds is at 250mA and that’s not counting driver or optics losses, assuming you are talking about Texas Ace’s results for the XP-L or XP-L2. Efficiency at 500mA or 700mA will be lower.
Even with a PWM-less linear driver like led4power’s you will only get 75-80% average driver efficiency at 500mA as the Vf is 2.8-2.9V, so the 0.8-0.9V to 3.7V is wasted as heat. I believe a buck driver capable of using a single cell like the LD-29 would get good efficiency if you could set the modes you like and lower the LVP voltage accordingly, but I’m not sure if there is a driver that makes that possible.
Looks like the trend nowadays is using a boost driver to run a quad-die LED like the XHP35 or the XHP50.2 so you can get high efficiency as each die runs at low current even at relatively high output. Using that approach Zebralight gets for example 14 hours at 196lm according to a review There is some technical discussion of boost drivers at [Buck and Boost Drivers, Testing, Modding, and Discussion]
Still, I think the cheapest way to solve your runtime problems would be training your subordinates to swap cells when the battery gets low rather than just leaving the worksite (if you are losing money because of short flashlight runtime as you described the problem is human, not technical).