Thanks, Robert! No python needed, folks. Not sure why this isn't flagged as the answer. Major kudos! Learn somethin' new every day 🙂
Just go into the Layout properties of your inset map (or in my case, a side panel map) and set the Display Constraint to "Fixed extent." I didn't have to specify a layer. I merely adjusted the map extent to where I wanted it beforehand, and then set the constraint afterwards.
Now, when the map series changes, the map doesn't move, but different features appear based on an attribute (forgot to mention, I also have a Page Query defined for the specific layer in the map that I want to display dynamically, based on the map series -- see below).
Set a Page Query on your layer and indicate what attribute needs to match the map series in order to be displayed.
Hope this helps someone!