The issue is that it doesn't work on our fully updated Windows 10 computers. But, the same package works perfect on Windows 7 with App-V Client verison 184.108.40.206. For testing, I upgraded the client on the Windows 7 machine to 5.1.134 and the package still works perfectly. I've still had no luck with the Windows 10 machines. The package was sequenced on Windows 10.
Watching process monitor, it doesn't even look at the "ComputerName" value I have set in the Virtual Registry. In Windows 7 it does.