Altium: Repeated Sheet in Hierarchical Design has Duplicate Sheet Number
I have solved the problem. There is a tool buried in Altium called "Annotate Compiled Sheets", found in Tools -> Annotation -> Annotate Compiled Sheets. In this dialog the user is allowed to manually enter Sheet Numbers for each of the sheets, including the repeated ones that do not show up in the "Projects" panel. 



Once these have been modified, in order to get it to print the page numbers correctly, in the Altium OutJob under the Schematic Prints configuration you must select "Sheet Number Parameter" under "Physical Name Expansion". This prints the correct page numbers.

In order to get the above to work, however, I had to get rid of the "Sheet 0" indicator and start the numbering at "Sheet 1" instead. Apparently Altium doesn't like starting at Sheet Number 0.

EDIT: For future reference regarding the following comment: 

Now I'm having the same problem with "DocumentNumber", except that Annotate Compiled Sheets doesn't have the Document Number as a parameter that is editable

If you are printing to a PDF using the "Smart PDF" tool, on the second-to-last pane ("Structure Settings") you'll need to check the "Use Physical Structure" box and check all of the other boxes except "Document Number Parameter". 

If you are printing to a PDF using an OutJob, after creating the PDF output, right-click it and select "Configure". Under "Physical Name Expansion" check all boxes except "Document Number Parameter". 

Provided the Document Number parameter field is set correctly, the above will remove the ".1"-".X" suffix on the end of the parameter, which was an issue I had with the sheet number at one point.