chances are good that it runs 3XAAA in SERIES , and that is very similar voltage to a li-ion.
the only problem being if the curcuit they use has problem around 3.6-3.8 as some of them would fall out of regulation around there, its not a problem, in fact its an assett, because the load drops when the capacity of the cells goes down. but that would be a concideration for things like wire length and resistance added because of the external device.
3 AAA have about 600-800ma total capacity when run with something like this, 3AAs would have at least 1900ma capacity under a load like that.
a Single 16500 would have about 1000ma and wedges in that space
a 18650 would have more than 2000ma (about 3x) and could be retrofitted or put in a pack. but wont fit easily in the space.
3x10400 AAA sized li-ions have no capacity and would be as worthless as the 3 AAAs that are in it, they would fit in the space but REQUIRE a full rewiring to parellel not series, and they are not protected, that would be a really useless thing to do.
if you wanted a all day cave light out of that thing, 4x18650 li-ion cells wired in parellel with protection on the pack or each cell, would be some 4000ma OR a single AW protected C cell at about 3000ma
do the math, a single or 2
protected C cells would make a great lightweight external li-ion pack, that would run for ever
then learn the safety, do you want a li-ion on your head?