SAT TRACON Controller expressed concern about airborne conflicts involving 5C1 airport; and suggested airspace redesign is necessary to enhance safety.

Narrative

Aircraft X was on the ILS approach to Runway 13R and cleared for the approach at the IAF. I noticed a VFR target flying west of 5C1 at approximately 032 just west of the fix QUESO. I issued traffic to another aircraft that was cleared at QUESO for the ILS approach. I continued to watch this VFR aircraft; ADSB identified as Aircraft Y; fly closer to the SAT Runway 13R final. I saw the VFR aircraft begin to appear to climb very slowly and encroach upon the 13R final. I decided to cancel the Aircraft X ILS approach and vector him away from this unknown traffic which was now tracking directly against the final for 13R and still climbing. I attempted to established radio contact in the blind with Aircraft Y; but there was no response. Shortly thereafter; Aircraft Y called to request permission into the Class C to overfly San Antonio downtown. I denied him this request because I did not feel confident the pilot understood the local airspace or could be trusted to fly safely after climbing directly into the face of a landing aircraft on a known and well established approach course to a major international airport. I vectored Aircraft X around the traffic for the approach to 13R.Once again I will reiterate; the 5C1 operation is a grave danger to the safety of the flying public. It is not if something will happen; it is when. There have been hundreds of reports concerning this busy uncontrolled airport that sits on a 15-mile final to SAT's main runway; but as of yet; despite many attempts the FAA has not acted. The FAA Manager who sits in a position to move this forward has changed multiple times over many years and each new person wants something new and different. We have done outreach to surrounding airports as requested; nothing has changed. We have done MORs but the FAA at the district level told us to stop doing it for reasons unknown. We have changed our procedures but there are limitations to how much we can mitigate the impending disaster that awaits the unfortunate souls aboard two aircraft in the future. The current FAA has asked us to consider reworking our arrival procedures again. The current arrivals were put in place after years of working with our airline partners to safely get aircraft into SAT.There are 2 options here. The first and most likely; is an extension to the Class C airspace that would prevent VFR aircraft from departing 5C1 towards SAT International at altitudes that are a grave danger to the safety of flight. The second is a Class B. I realize there is more to a Class B than just safety; but it is still an option.There is a 3rd option and at this point it seems the FAA is content to continue going with this. That is do nothing. For over a decade; controllers have been sounding the alarm about this and the FAA seems perfectly content to do nothing. With all sincerity; I truly hope the FAA changes course here with us to prevent more loss of life within the NAS.