Yeah fuck that. US$13 on Amazon - Pollak 11-893P RV 7-Way Socket... https://www.amazon.com/dp/B000B6NQHW
My first reaction is not to trust this dealership. I have the AWD Pro S non-Gradient. I have had zero issues with the car since October.
I can confirm that the towing wiring harness is factory installed and your electrical installer is telling the truth. I ordered the Pollak 7-way socket (https://www.amazon.com/gp/product/B000B6NQHW/ref=ppx_od_dt_b_asin_title_s00?ie=UTF8&th=1) and installed it myself. I haven't tested it by hooking up trailer lights to it though.
As far as the FitCamX, I don't really buy that. There are multiple owners on vwidtalk that have installed that model dashcam and had no issues. I installed a BlackVue dashcam and wired it into the always-on overhead console power so it's active 24/7.
I'm at a loss for advice to give. I guess have them show you the "aftermarket module" before they take the bumper off and see if there is any truth to that claim.
EDIT: I read your post again and it's not clear who is servicing your car. You are at a VW dealer correct?
I purchased this adapter on Amazon for $15 and it has 7 Blade to 6 Round, 5 & 4 Flat options. Have used it with a Home Depot utility trailer that required a 4 flat.
Assuming you already changed the European plug to a 7 blade which I did with this 7 blade adapter from Amazon which is the same as OEM but 70% less
Here is a link for the 7 pin socket. The dealer will sell you the same thing for about $55, or you can order it here for about $15. https://www.amazon.com/gp/product/B000B6NQHW/ref=ppx_yo_dt_b_asin_image_o05_s00?ie=UTF8&th=1
Pollak 11-893P