I rubber ducked through it and got it working by doing the following:
Change the initial value of my INITIAL_FACILITY_SELECTION to blank, and create a JS query that runs on page load to set the variable's value. Not sure why this is any different than setting the initial value field on the variable, though.
I had to set a delay on the success event handler for my dashboard query that sets the local storage value to avoid any race conditions between setting the initial value of the dropdown and setting the selected value in local storage.
And for extra safety, I set the dashboard query's disable criteria to check to make sure a value has been selected so that it doesn't run while the initial value input is being set.
Sorry for wasting anyone's time, but sometimes you figure stuff out when you type out everything you've tried :-). Hopefully this will help someone in the future.