So following up here:
BLUF: I changed the alternator and the issue is resolved.
After my original post, I let the vehicle sit for 8+ hours and the problem didn't come up again for several days. But it did present eventually. We were actually on the road pretty far from town, and after a fill up, I turned the car on and got the warning. Tried revving it to get past it as before, but no luck. Checked the battery voltage and it was steadily dropping despite any RPM changes (especially with electrical systems running). I shut it off about 5-10 mins while deciding what to do (again, far from home), turned it on again, revved it, and got 14.5 volts again... Since we had a full tank (and 190 miles to go), I didn't shut it down again until I got us where we were going, dropped the wife off, picked up a Duralast Gold alternator and went to my friend's house in town to do the swap. I didn't know for sure if that would fix it, and I DEFINITELY wanted an OEM part, but I needed to try and OEM wasn't available on my time line, so faith in Duralast it was. 4 hours later we had the Edge back together and haven't had any issues since. Voltage is steady at 14.7 now upon startup and we're well over 1k miles since (currently still on the way home).
The original alternator terminals were severely corroded which I couldn't really see until we removed it, and the corrosion seemed to go "deep".
We basically did everything in the video linked below (which I'm sure many here have already seen). One observation though is that loosening the belt wasn't as easy as we thought. There's very little room to get a tool in there, so we used a crowbar and a large impact socket to offset it from the engine mount for leverage. I pried it back while my buddy pulled the belt off / put a new one back on.