Update: So I seem to have figured out that Launchbox is indeed somehow messing with how Retroarch is interacting with Retroachievements, or at least with how Retroarch is launching. I've attached two video files to demonstrate what I'm talking about.
The first is merely me launching Retroarch from my desktop shortcut and quick selecting the last game on my list. It happens to be Banjo-Kazooie for the sake of this demonstration. You'll notice that the widget popups occur as normal, and it successfully manages to load achievements.
The second video is me doing the exact same action (launching Retroarch from my desktop shortcut), however this time, I have Launchbox in the background. You can clearly see that the widget popups look different, and that it fails to load up those achievements. Therefore I can only conclude that something inside of Launchbox is interfering with this function, as its the only changed variable between the two launches.
What baffles me is how or why its doing this. As I've said before, I've got a single installation of Retroarch and a single installation of Launchbox. The directories are all lined up, and in fact have no changed in the 2-3 years I've been using this program (which I'd like to add is amazing and I've had zero other issues with thus far). I've tried removing and re-adding the emulator, I've tried tinkering with the executable, and I've even completely rebuilt the Retroarch configuration files, and nothing had changed this launch pattern.
Anyway, I'll continue to look into this. Thanks for help thus far!
Retroarch Alone.mkv
Retroarch with Launchbox Open.mkv