I have an F1 which locks with mirror up and shutter open on the slow speeds- 1/8 sec and lower. I suspect the slow speed governor needs a clean and oil. Can I get to the governor without removing the film counter unit and/or shutter speed selector? The governor is under there somewhere but the Canon service manual is not helpful. Removing the units looks tricky. I have the same problem with a Pellix and probably the same solution too.
You need to remove the winding/ counter plate. This is easy since the screws are positioning screws and alow only one location. The only trick is to pull the counter away from the doow latch in the front. Also remove the sync wire by unsoldering it. This should always be done on these cameras becaus of the age and deterrioraton of the foam insulator. The governor is there and not too easily serviced without pulling it which requires the removal of the shutter speed plate. However since this is just for your usage you aren't interested in factory specifications so try to do a simple flush cleaning. Be careful of the leather used for the breaks that you don't get cleaner on them. Also check them while you have the winding plate out. Watch for the location of the sync switch when you reinstall the plate.