As a follow up to this (in case anyone searching here ever experiences the same issue,) the culprit was indeed the input jack, as suggested by Tim. However, the issue wasn't with the switch clearing after taking out my instrument cable...
After a bit more prodding as suggested by PeterJ in the comments:
@TimBurnett-Bassist you won't get a reading at the jack because it's
switching the negative / ground line. Put the positive of the
multimeter on the red wire where the battery is spliced and put the
negative to the green followed by the black wires on the jack, if you
get a reading on both with it unplugged the circuit will be getting
power all the time because the internal switch is stuck. – PeterJ Nov
4 at 10:39
I noticed that part of the jack was actually touching the bottom of the pot just behind it, thereby creating the short:
@PeterJ - I just did what you suggested and I was getting a reading on
both the green and black wires on the jack. However, I noticed that
the green wire and its terminal were touching the pot that is just
behind it... I twisted the wire and terminal away from the pot and now
I get a reading from only the green wire, not the black – Tim Burnett
- Bassist Nov 4 at 16:51

Since I've pulled the jack away from the pot so it's no longer touching, I have had no issues whatsoever with the battery draining incessantly. Thanks for all the help, folks!