If I follow correctly, your test was to put smaller size tires on your jeep, after which it did not have any false activation of stability control. This seems to suggest that your Jeep may indeed be programmed for the stock size tires... If this was the case, then running 35's would make the stability control more likely to falsely detect an oversteer condition when taking sharper turns, particularly up near the limits of comfort, where the threasholds for activation are tighter. (Banked turns on tight curves are tough areas for stability control calibration)
Did you get to double checking the procal settings? That would be the first thing I would do, then verify that the speedo is reading a mph or two faster than you are actually going at 65mph. (Google maps will give your speed if you have it give you directions, and it's pretty accurate.)
If the speedo is indeed accurate, then the tires size programming should be correct. If it reads exactly right, or slower than your actual speed, you need to update the settings to a larger tire size so that it reads just a little high. This would mimic the factory calibration, which is usually a mph or two fast at 65mph.
The next thing I would check is all of your sway bar end links... If you are getting excessive body roll in turns, it can cause a bit of roll steer. That would look like additional yaw vs. what steering angle alone would produce, and would cause you to have some sensitivity, as you do. With aftermarket shocks, you probably wouldn't notice in day to day driving. (Hmm, it was a common failure on TJ's to have the ball stud pulling out, but on JK they fixed it so the stud is 90° to the link, rather than aligned... You should get clunking long before any complete separation on your JK...) How about a bunch of weight on the roof? That'll cause more body roll too.
Perhaps your "issue" is multiple things all contributing a little error... a little speed error due to tire size, plus a little error due to the increase in track width, plus a bit more body roll in turns due to whatever... Running a smaller tire than you are programmed for would cover this up nicely, which is what you saw.
With everything else being OK, you could try removing the spacers and see if that makes enough difference to eliminate the false activations...
I would NOT suspect tire choice would make that much difference unless you hear howling in otherwise comfortable turns. I had some Michelin tires that were just awful long ago, and would howl and slide at posted on-ramp speeds. If you're not getting that, you should have no tire issues. Your pressures seem good for street driving, and sidewall deflection at comfortable speeds is not significant enough to cause issues, IMO. Worst case should actually be a mud tire, with tall squirmy tread blocks, while you report no issues with previous mud tires, but issues running AT's of the same size. That makes no sense, so I don't think it's a factor.
In short, something has changed if you were not getting this before. You said your procal was stolen and replaced. That's a change, and you can check it. Did you add spacers when you put the AT's on, or were they there with the MT's too? Another change perhaps... Aftermarket sway bar would affect things some, worn out links with some slop would be something that would get worse over time...
