You can get a relative measurement with a photodiode or photoresistor, yes. You need to calibrate at a few points (photodiode) or most of the curve (photoresistor), and preferably it should be done in a dark enclosure with a consistent optical setup (everything clamped, etc.).
If you have access to the internals, you can replace the LED with a schottky diode (if its supply is AC, often the case for boost type circuits) and filter cap, and connect an electronic load to emulate an LED while measuring the DC power output. If it's switching, it shouldn't be far off to measure the input power just as well, and assume a modest efficiency (80-95%?).
Should be pretty obvious in any case, once you start varying Vin.
Incidentally, about switchers -- everything should stop when the cell hits 3.0V or so, as further discharge is hazardous. The control may simply shut off, in which case, the voltage then slowly recovers, and then it flashes again, and so on. Or it may throttle down, effectively giving you a flashlight that dims like an unregulated one does, but all the dimming is squished towards the fully-discharged end, not gradual throughout the charge curve.
Tim