If I had previously "archived" a dog and they reach out for a different booking, how to I relocate their request out of the archived section and into my pending section?

Huh. It's kind of design flaw that you can't do it in the mobile app IMO.

You can also find the Archived and Past booking in the App. It's on the same page of Inbox, press the up left corner there is a folder icon. Then you can see all the past and archived booking.

The booking should be in your archive folder, you need to go to the webpage, not the App to find them, once you do you can unarchive the booking request with a simple click.