SAT TRACON Controller reported a conflict developed between a VFR aircraft operating at 5C1 and a commercial jet. Reporter recommended changes to the SAT airspace structure to prevent future occurrences.

Narrative

I was working the standard combined radar position. There were multiple medevac flights inbound and I was trying to ensure no delay for them at the time. That meant that Aircraft X was slightly extended to ensure the medevac ahead of him had no delay on their ILS. I turned Aircraft X base and descended them to 030 feet planning on the ILS to RWY 13R. There was a VFR aircraft working in the 5C1 pattern throughout the evening so when I saw a VFR target departing 5C1 I didn't think anything of it. Then the aircraft departing 5C1 turned left and began an aggressive climb (<1000ft/min) through 2;200 requesting an IFR clearance they had filed. I began the process of radar identifying the aircraft while simultaneously using the mult-d button to determine if the aircraft departing 5C1 was the same one calling. I immediately began traffic alert procedures because Aircraft Y was climbing into Aircraft X. Aircraft Y was instructed to maintain VFR and Aircraft X was instructed to maintain 3;500 and turned to 160 to avoid. Aircraft Y went up through 3;000 and Aircraft X advised that the opposing traffic flew right under him and he was responding to a TCAS RA.Suggestions: This event is not uncommon at SAT. In fact; this report is one of many associated with traffic in and around 5C1 and conflicting SAT traffic. I have said this before and I will repeat it here. It is not a matter of IF there will be a disaster at 5C1 it is a matter of WHEN. As a facility we have jumped through every hoop the FAA has presented us with to try and prevent dangerous situations from occurring. From outreach at the location (many pilots do not attend) to tracking the problem through the MOR process. So at this point; I can only say unless the FAA wants another DCA disaster to occur; they will act decisively to fix this problem.The answer to the issue is very simple. SAT needs controlled airspace to prevent aircraft from working east of 5C1. This could be an extension to our Class C to encompass an area just east of 5C1 or a Class B to do the same. The point is; relying on pilots to operate under VFR see and avoid with no knowledge from the controllers as to what they will or will not do is completely unsafe.