FYI, a CR123A isn't 3.7V, even open circuit. OCV is closer to 3.2V. Under a 1A load, it is probably closer to 2.5V-2.75V.
Does that driver use the usual AMC7135 LDO regulator? If so, I see 8 of them, which would mean 2.8A out. An IMR could help. Or the reason the board outputs 2.5A instead of 2.8A could be that the PWR controller IC is programmed that way for High mode. Or the board uses a different LDO regulator. Or it's the usual lousy spec'ing of these cheap Chinese drivers.
Assuming an 8xAMC7135-based driver, Vbatt > Vf + AMC7135 drop (=0.12V) for the board to run in full regulation. Let's assume Vf=3.50V (the high end of an I voltage bin P7 and the low end of a J voltage bin P7). Then, Vbatt > ~3.62V. Should be easily achievable using one Li-ion for most of its run time. However, if you are unlucky, and the P7 Vf is 3.75V (high end for a J-bin), then Vbatt > ~3.9V. Now, voltage sag could get you and an IMR, with its lower internal resistance, could help.
If you can put your DMM probes across the P7's terminals while under load to measure Vf, that would help narrow the possibilities.