After much communication with Squarespace Support (they were very patient and helpful via email), the ultimate fix was:
In Facebook > Settings and Privacy > Settings > Business integrations, beside Squarespace, click Remove.
Log into Squarespace and navigate to the Settings > Connected accounts panel.
Click the Facebook account, then click Disconnect and Confirm.
Re-connect the Facebook account https://support.squarespace.com/hc/en-us/articles/206544007-Connecting-Facebook-to-Squarespace
Click Okay for each of the permissions questions you're asked, if you are comfortable with the permissions that are needed to connect your Facebook page to your Squarespace site.
When you're done, click the newly re-connected Facebook account again and see if your page appears in the drop-down.
Even though I had tried this many times, this last time seemed to fix the issue.
Good luck, hope this helps you as well!