You did mention change when stopped. I.E. under no load, free running.
That ,to me, suggests then something else like a harmonic vibration. Try different RPM's and that should confirm.
Often it would be a large surface (like mower deck, cab roof etc) the is the actual cause.
If U confirm that sort of thing then dampening the actual generator is rather easy.
Like adding a stiffener, gluing strips of carpet even adhering some self adhesive roofing membrane will do the trick.
On my DIY cab I anticipated that effect so I adhered some deadening material under my roof, no problems.