Can I restore a differential backup to the 2nd newest full backup? At 12:30 pm we pulled a full backup and shipped it, and set all users in read-only mode. At 11pm that night another full backup was triggered.

Assuming no changes occurred during that time span, at least none that I care about, can I apply a differential I pull now to the [restored] 12:30pm backup I created...or will it not work/error out because another full backup occurred since it?
No, that's not possible. You will not be able to start the backup operation.

What you are trying to do there could leave the database physically inconsistent if SQL Server allowed it. You would get a Frankenstein database with pages mixed from different points in time.