I have this all working now.
After much digging I found this: How to do AsyncPostBackTrigger for the LinkButton in the Repeater
I was able to use this in my Page directive: ClientIDMode="AutoID"
I also ended up adding an ImageButton to my Repeater datarow, and whether I click on the LinkButton, ImageButton, or Search button I get the desired result now - the UpdateProgress fires and displays the animated .gif file.
My UpdateProgress resides inside of my UpdatePanel and has an AssociatedUpdatePanelID set to the ID of my UpdatePanel.
I hope this helps someone.