Answers: You're not making satisfactory power to run the engine and the lights at the same time - this sounds resembling the battery or the alternator. I know you said that they be both tested, but if the battery have an internal break (usually caused by overtightening the terminal, or sharp impacts), it can test fine and still not work when within the car - my wife's vehicle did this to us last winter. The solely other possibility I can think of is some sort of unusual electrical glitch - and finding those can be very time consuming - and expensive.
Good luck!
Might hold a ground issue something to do with the headlights.
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