Electrical problems can be really frustrating. I am sorry you are having these issues.
Replacement 3rd brake lights are all over the place. If you suspect a problem with it, why not get something different (something that you like better than the stock offering)?
Replacing a bad part would be the perfect time to upgrade.
You've already done most of the work, anyway.
For my Ranger (CX Dark Shadow Grey Metallic), I wanted something that blend in better with the cab (until it came on). I got
https://www.carid.com/spyder/chrome-...97&url=3316951
It wasn't just plug-n-play, but I figured it out.
This unit (the one I got, anyway) had assembly screws that crashed the sheet metal under the brake light secure grommets, so the unit wouldn't sit flush and seal to the cab. (That is a water intrusion point when the 3rd light doesn't seal.) But when I took the screws out, the unit wouldn't come apart. Those screws must have been there to hold the thing together while the glue set (sonic welding - whatever process sealed it.) I tossed the redundant 3rd light assembly screws and Dremeled down the screw pads. Then the unit fit right. Since installing it, I've had no problems, and that was nearly 4 years ago.