Spot on - I made sure my PCSX2 had -batch as well as --fullscreen. I have since removed both command lines and changed the settings within PCSX2 Nightly itself to "Launch in Fullscreen," which works just as well.
Since I made this topic, I decided to dive into a couple of prior topics from a couple years ago on these forums. I found the solution!
In hopes that this helps someone else, MAKE SURE TO CHANGE YOUR CONTROLLER MAPPINGS WITHIN BIG BOX, NOT LAUNCH BOX! This little detail tripped me up at first.
The next detail that I failed to notice was that the hotkey I needed to map was for "Close the Active Window," NOT "Exit."
Reference screenshot:
Note that "Close the Active Window" hotkey is way down towards the bottom. After changing these settings and re-mapping the "Pause Emulation" button within PCSX2 itself, I finally have a fully-working Big Box exactly as I wanted.
Thank you so much!!