IF SIXT file a claim. You do NOT file a claim until SIXT charge you the damage. I don't think SIXT would file a claim directly with your insurance company. Not even US rental companies aboard. I once had a damage with Budget at Sicily. At car return it immediately charged my CC, with a detailed damage repair invoice together with 3 pictures, sent to my email real time. I then spent half an hour at the counter to get the Rental Clising Agreement for the final details.
Then once returning home I filed an eclaim with my CC which was Chase Saphire Reserve. The claim was settled within 7 days from filing to receiving approval with instruction on which payment option I wanted. There were some uncovered items such as Airport Surcharge on repair. All in all, out of 1300+ euro invoice, CC CDW paid almost 1200 euro. Our out of pocket iirc was about 130 euro.
I am confident you would have similar outcome if it wound up has to be a claim with your insurance.
At this stage you can continue to dispute with SIXT, though given its reputation, you might not succeed.
Good luck.
