I came now, late to the game.
The LIC is a hybrid between LiPo and EDLC, Polymer Caps. So low ESR for higher power density and wider discharge voltage range than a battery but lower range energy capacity.
The under-voltage UVP protection threshold depends on your life cycle expectations, so choose conservatively or wisely as short-term energy gains cost dearly in the long run. That same is true for Lithium batteries so you could get 10x the lifetime Wh use if you only used 50% capacity each charge by reducing time over 3.7V and cutout at a higher voltage. Your choice of cutoff hysteresis value must exceed that of the ESR*I range of the storage cap/battery to prevent rapid oscillation This can be improved by a slow cutoff..
A low-side Nch power switch with a stable Vref is a minimal solution for a "hard" cutout which is the DO discharge output side of the battery protect IC. If you prefer a "soft cutout" that slows down before shutting off then a PWM-controlled DO-like switch comparing Vout to Vref cutout range could be designed,
All it takes is more specific design specs. with special attention to the resistance of every part of the design.