a few things.
I've powered a 328 arduino chip from a bare lipo, unregulated, just taking its output direct, and it runs until it gets to 2.8v or so. this is a 'normal' 328p chip using a 16mhz resonator, no special bootloader setting fuses to non-5v/16mhz states. I was surprised it went down that low. note, it was a bare chip and resonator on perf board, not a prepared green board from a vendor.
also, make sure you have a lvc (low voltage cutoff) SOMEWHERE. MANY batteries do not have it anymore! all aircraft batteries for toy quads, for example, seem to omit the cutoff circuit and require the copter to have the lvc in its logic, not the battery. for 'land batteries' I've heard that the circuit IS in the battery. so, YMMV and you need to know this. its BAD to undervolt a lipo, as I'm sure you know.