I sense that this thread is drifting from the OP's topic more towards specifically incandescent-related options instead of LED-related options.
Which is fine, but all of the above discussion has been covered much more extensively in that subforum.
I'll link up the "rechargeable options" sticky below.
The second issue is that battery terminology is very confusing, and that lamps are much more unforgiving on those specifics.
It makes a great deal of difference if a "rechargeable CR123" is IMR, ICR, IFR, protected, unprotected, 3.0V-diode, 3.6V, 3.7V, 4.2V, 4.35V, etc, etc, etc.