Being an SRT, there are more issues that can cause this than in the N/A. At the mileage of your car, a new Intercooler Pump may be in order. (Not a hard fix, and the pumps are not expensive at all.)

Given this "runs ok cold" stuff, I'd say I/C pump or Crank Sensor. Often, the RIGHT sensor is less money than the "cheap" ones...
https://www.germanautosupply.com/sto...sensor-82.html How does $45 grab ya?

This thing about the tow alarm light flashing has me wondering - how old is the battery? These cars are very fussy, at times, about the battery. (And PLEASE don't tell me it "tests fine" - I didn't ask if it 'tests fine', I asked how old it is. The issue is not starting amperage, but filtering ability - again, the car is not tolerant of voltage fluctuations or spikes - and it is the battery's job to filter them out.)