I think more likely than your own insurance company denying a claim due to exceeding GVWR is being on the far end of a lawsuit by another party in the accident. Your insurance company would be obligated to cover you up to the policy limits, but you could very well be held liable (even criminally liable) if you were knowingly driving an overweight vehicle.
Remember that a lawsuit is a very blunt instrument. Even being right doesn't keep you from being sued, and once in court very little is black and white. They would claim you were exceeding the vehicle's capacity and were therefore de facto negligent, you would be claiming that due to axle ratings or upgraded tires or whatever you weren't. You are just hoping your expensive lawyer is more persuasive than their expensive lawyer. It becomes almost a******shoot.