I rode a beater with an inexpensive stamped suicide hub for many years with a fixed cog installed. I did use red Loktite, but didn't bother with a secondary nut.

The only time the fixed cog ever loosened was a day when I deliberately, actively worked on skids. Track standing or braking by back pressure on the pedals had no ill affect on the installation. Keep both brakes installed and you'll be fine unless you're bent on destroyimg a rear tire.