To get FPM to work with MO or F4SE, set Monitoring Mode to "Option 2".  This will tell FPM to wait until it detects Fallout 4 running, and then being monitoring.

Steps:

• Start FPM.
• In the Tools menu, click "Setup Window...".
• Click on the "Miscellaneous Settings" Tab.
• Set the "Monitoring Mode" to "Option 2".
• Click the "Save" button.
• Back on the main FPM Window, click the "Monitor Fallout 4" button.
• FPM will now begin waiting for Fallout 4 to be launched, and the Status Field should now read "Waiting for Fallout4.exe".
• Launch Fallout 4 using Mod Organizer, F4SE, Steam, etc.
• FPM should detect Fallout4.exe running, and begin monitoring.