LED tail lights on a flat bed

I suspect this issue is the same as the turn signals. You have a stop lamp relay that powers the stop lamps and it runs through the turn signal switch.

So I suspect it is the relay and not the switch but it may be both as LED lights have far less resistance then the halogen bulbs and it throws off the relays and control modules if it has them.

I would suggest checking the voltage from the relay on the highlighted wire from the relay

https://www.2carpros.com/articles/how-to-check-wiring

https://www.2carpros.com/articles/how-to-use-a-test-light-circuit-tester

I attached the wiring diagram below to assist with this.

Please let me know what questions you have and we can go from there. Thanks