Accident shopping cart, holder not equal driver – priv. liability or KfZ liability?

Hello everybody! Last Saturday I passed the following painter in the (hilly) parking lot of a shopping center: I bought a large carpet and transported it to my car with a shopping car. first I put it sideways to my car, but had to take it away, because another car just wanted to park me. anyway I took the carpet out of the shopping car. in this eye view this sat in motion and rammed a parking peugeot in the rear wing, which left behind three small dellen. since I already have one or the other delle from the parking lot and know how annoying it is and also the opinion is that one has to stand for the crap that one also has to read, I had the driver of the car exclaimed, although bystanders thought that there was nothing to be seen (^^). Well, on Monday the accident opponent called me and said his workshop had caused a damage of 732€, since no lacquers If you have any problems with your car, you will have to ask me if you have any problems with your car, but the bike box got a little bit off… (it went out from about 50-100€). Now my question is: the car which I drove is mine (owner according to §334ff BGB), i.e. I have approved the letter and the KV, but it is insured on the mother of my girlfriend. now I have of course reported the damage to my private liability insurance (at first by telephone). Does the private HV or the KfZ-HV for the damage? Thank you for your help and LG Ka rlsquell78