Solution
This can be achieved by launching LabVIEW from the command line using the -pref flag, and specifying the custom INI file. For LabVIEW 2020 64-bit, use a command similar to the following from a Windows Command Prompt:
"C:\Program Files\National Instruments\LabVIEW 2020\LabVIEW.exe" -pref "C:\MyCustomConfig\labview.ini"
The flag for the INI file can point anywhere the user places it. On Linux, a command similar to the following can be entered in a terminal emulator:
/usr/local/natinst/LabVIEW-2020-64/labview -pref /directory/labview.conf