Not a lawyer here but definitely would drop them back an email explaining the issue.
Being a foreign national you can easily prove that the lack of reply about the actual driver occurred in good faith, since no additional payment was due anyway by your side, once the original fine got paid.

Better to write them in Italian. If nobody can help you PM me and will be glad to help with the translation.

Did you get your second request via Hertz, or directly from the Police Dept.?
This makes a big difference in terms of collection capability by the road body.

As a side note, not sure if unpaid fines get channeled to immigration department, but my guess is that it is unlikely.
For 310 euros I'd definitely fight back, and simply carry the receipt on the next visit (not necessary anyway after 5 years from the infringement).