I'm thinking of reinstalling and only applying the bring down the sky addon and not the patch.
You can not do that. The patch is required and included in the installer for BDtS. You should not have installed the standalone patch if you already installed BDtS - if you did that, it could possibly be the cause of your problems. The standalone patch should be used only if you want to patch the game without installing BDtS.
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The sticking overheat thing was intentionally inserted as an anti-piracy deterrent. Apparently the devs put in several little things like that designed to degrade the overall experience so that even if the initial authentication check for starting the game was cracked, a pirated copy of the game would still be problematic.
That's not to say that I'm suggesting you have a pirated copy. Rather perhaps there was some authentication failure as a result of the patch, or upgrading your hardware or something like that, and now it's behaving like an illegitimate copy. There are plenty of legitimate owners that have reported the overheating problem. That's the risk with these traps and why I dislike them. They're clever in theory, but in practical reality it's too easy for something to go wrong and screw up legitimate copies of the game.