Not true at all. You fucked your Carfax report up when the police got involved. When there’s a police report, it definitely goes on Carfax. When there’s an incident with no police involved, it usually doesn’t. I got in an accident with my G35, and got it repaired through insurance but it never showed on the Carfax when I traded the car in. Reason for that is no police report.
Might as well claim it on your insurance now anyways, you will barely be paying much more than you are now, and since there’s a police report and you’re pressing charges, the increase will be insignificant or none at all.