This happens all the time in the parking lots, especially when the shopping cart is full of goods. The lady park the cart near her car, so she can open the trunk. The cart slid and hit your car. This is an accident out of nature. You are talking about three dents on your bumper. How big are the dents or just a scratch?
All cars that sell in the US, its bumper have to withstand 30 mile per hr direct hit without any damage.
It’s very hard for you to prove that dents are cause by the shopping cart.
You can report to your Car insurance, but I am very doubts, they are going to replace your bumper.
In the court room, you have to prove how the cart gave you three dents instead one.
I feel bad for you, but you are not going to get anywhere.
No- you premium would not go up.