HPN Local Controller reported RA response with non communicating traffic resulted in an low altitude alert and CFTT event. Controller also reported the traffic volume exceeds requirements for Class C which will prevent these incidents.

Narrative

While I was working LC (Local Control) and CIC (Controller in Charge) and was moderately busy with IFR and VFR traffic. I was focused on getting departures out as well as trying to separate and sequence a VFR Aircraft X and a VFR Aircraft Y from the west. The first time I heard Aircraft Z check in was when they were on a 5NM final with the conflict alert going off. There was a VFR target less than a mile northeast of their position. I could not read the altitude of the VFR intruder due to tag overlap. I issued traffic; and Aircraft Z advised they were responding to an RA. They descended; triggering a low altitude alert; which I issued. I cleared Aircraft Z to land; and they landed Runway 16 without further incident. After listening to the recording; it seems Aircraft Z tried to check in about a 50 seconds prior to the incident; but I must have been distracted or did not hear them at the time. A Class C shelf may have pushed the VFR target to communicate with ATC prior to it becoming an issue. HPN greatly exceeds the traffic and airline passenger volume requirements for Class C. An upgrade to an HPN Class C airspace has been in the approval process for the past few years. Recently talks have stalled with N90. I think we should push to make this change as soon as practical.