Strong Shaking/Vibrating Below Center Console at Low-Speed Hard Acceleration

I went to test drive an i4 M50 yesterday which I would trade-in my S5 for so I let the dealer take it for a quick drive to make sure everything was in order. I had been driving my S5 regularly and fairly aggressively the past few weeks and everything was perfect but as soon as he kicked down on the accelerator a very violent shaking/vibrating started to happen right below the armrest maybe a bit to the back. My sister in the rear seats said it felt worse there.

As this wasn't an issue I had noticed or heard of before it put me in a pretty crappy position to negotiate the trade-in but honestly this is the last straw for me with the moneypit this car has been. However I am curious if this is something that sounds familiar to any of you and if it's more or less than 4k to fix? Because that's the hit I took on the trade-in since the dealer also has a shop where they'll try to fix the issue and I might get up to 1k back once they figure it out and send me a listing of the parts they needed to replace.

I'm wondering if getting an indie mechanic to fix it would be the smarter move since it could "just" be transmission mounts or the driveshaft but we also put the car on their lift and couldn't see any visible shaking.