Momentary overloading of a mains transformer at low duty cycle operation - what could go wrong?
I believe you have a handle on it.   P = I^2 * R.  The thermal rise in a resistive device will be the square of current.  That puts the square in  RMS.  

This is why RMS has the relationship it does with peak.  This is the AC voltage that, when applied to an incandescent light, will provide the same light, heat and service life as that DC voltage.