It should be easy enough to just start Firefox and hit Ctrl+Shift+P. The official documentation on command line options highlights that running firefox -private should get you into private mode easily. Also there should be support for -private-window and -private-window <URL>.
Shamelessly stealing from my answer on StackOverflow:
Links to local files on pages that were retrieved via HTTP(S) are deliberately disabled in Mozilla/Firefox, because they can be a security risk, and have been used in attacks in the past.
You can override this behaviour, however.
You need to configure Firefox' Security Policy to allow ...
For quick reference-
Firefox Advanced Options
You can enable the auto-refresh from Firefox Advanced Options by clicking the orange Firefox button on the top left corner of the browser and then clicking "Options." From the list of options, select "Advanced." Under the "General" tab, in the "Accessibility" ...
I implemented what I need with the NppExec plugin. After installing it, I opened its Execute dialog and wrote the following script:
"C:\Program Files\Mozilla Firefox\firefox.exe" "$(FULL_CURRENT_PATH)"
Then saved it as a macro with the name 'Save and Open in Firefox'. After this I opened the Advanced Options for the NppExec plugin ...