This helpful question pointed me into the right direction for powering the BBB with a 3.7V LiPo battery. However, as this answer demonstrated, I won't be able to give the WiFi/BT module power in doing so.
User rblasco mentioned that placing a STPS1L30U diode solved this, but why was this diode removed from the BBBW in the first place? Would there be any negative ramifications if I also added the missing STPS140A diode?
Edit I've found out that I can get the BBBW to boot with a LiPo battery. This question is asking if anything bad happens from applying the modification.