Paying cash for fender bender rather than going through insurance

At what point does it make sense to pay cash for a fender bender instead of going through insurance?

I was backing into a parking spot when a passenger in the vehicle in the next spot opened their door and I backed into it. No damage to my bumper but damage to door. I asked the owner to get an estimate from a body shop and it came out to ~$4500.

My premium for six months for my vehicle and my partner's vehicle is only $2165. It seems like I should tell the other driver to go ahead and claim on the insurance.

Is there anything I am overlooking?