Database Properties Error - sys.spt_values missing but dbo.spt_values works?
Answer compiled from comments by the question author

Following a suggestion by Ben Thul to try connecting using the Dedicated Administrator Connection (DAC), that was it: I was able to connect over the DAC wtih SQLCMD and run

select * from sys.spt_values


From there I ran the u_Tables.sql script and it looks like we're good to go.

I'm not familiar everything that is different when using the DAC, but that connection allowed me to see the sys.spt_values object.  I had not been otherwise able to query against it.  Since the object was now visible over the DAC, I could execute the u_tables.sql rebuild script successfully.

It is still unclear what caused the original issue. I was contacted by a client regarding the error in our Test environment. The issue did not impact any other functionality besides the properties dialog box in SSMS.

Several sources said to run the u_tables.sql script to fix the issues. When I inspected the script, I tried to run just the sql within the view and kept getting the error about an invalid object.

This was true regardless of how I connected through SSMS. It also meant the u_tables.sql would not execute successfully.  The DAC connection allowed me to see the necessary objects.