hello welcome to the site.
does the dash lights act up when your starting the car ?
when started can you wiggle the key in between the start and the II position ?
thank you for the update.
since this happened under full load i would recommend you have the alternator tested with all the loads on like the heater and the headlights.
im thinking the alternator is on its way out.
or you might have a ignition switch electrical portion going bad.
let me know what the test results are.