A TRACON Controller reported they vectored an IFR aircraft to avoid a VFR aircraft not in communication. When the IFR aircraft responded to a TCAS/RA it descended below the MVA.

Narrative

Aircraft Y was not on the frequency; flying level and consistently on a heading of approximately 310. Aircraft X was on vectors for the Visual Approach; and I identified the conflict about 7 miles apart and converging. I issued a vector away from Aircraft Y to Aircraft X to avert the conflict. Aircraft Y then started turning north into Aircraft X and I was unsure of what to do because of the unexpected turn and limiting airspace. Aircraft Y was observed starting a descent; and I issued a climb to 14;000 ft. per the Supervisor's instructions while they were 2.5 miles apart. Aircraft Y then started a rapid climb and another turn directly into Aircraft X from only 2 miles away. Aircraft X reported an RA descent; which put them 300 ft. below the MVA in the area. I issued a low altitude alert. and when clear of RA; Aircraft X was then vectored back for the approach.Class Charlie Airspace could have been very advantageous. From the replays; it appeared Aircraft Y may have saw Aircraft X and may working to avoid them. However the unexpected changed of course; and the unexpected climb put the airplane in unsafe proximity. It also resulted an an IFR aircraft being put at an unsafe altitude. If Aircraft Y was on the frequency; I could have known the intentions long before they flew through a bad spot on final; and provided minimal vectoring to ensure a safe and efficient flow of traffic for both aircraft.