If the sound is really important then you should get a sound guy. Maybe someone pointing a shotgun to the speakers and adjusting the levels in a pre before you get the signal to your cam.
Some handy-cams have an auto adjustable level for the sound. If you have one of those, get a friend to get the piece in a stable position while you record the main footage.
I know you were asking for a single button solution, but good sound almost always require a little extra investment both in hardware and personnel. Auto adjustable sound level doesn’t come but in cheap cameras and there’s a reason for that, it doesn’t always sound good. Convince of that to the people that hired you.
I hope this helps
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