Why can't I select my grouped video clip to reposition it within the frame in Premiere Pro?
The video clip I'm trying to reposition has been split into three separate clips with the Cutting Tool and then Grouped together.


This is the key to your problem (emphasis mine). 

In Premiere Pro, you cannot move a video clip around within the frame in your Program Monitor view if it has been grouped together with other timeline objects. The application would have to reposition each grouped item with relation to the clip you are trying to select and reposition. This kind of chained action simply isn't supported, so it just prevents you from selecting the clip within the Program Monitor to begin with.

You will have to select each of the grouped video clips on your Timeline, Ctrl-Click to bring up the context menu, and then select Ungroup:



You will then be able to reposition your video clips within the frame. You can re-Group the video clips when you are finished, if you like.