A SAT TRACON Controller reported they had to turn and cancel descent for an aircraft cleared for an ILS approach to avoid an unidentified VFR aircraft crossing the approach course.

Narrative

Aircraft X was inbound for runway 13R at SAT and was cleared for the ILS approach. When Aircraft X reached approximately 7000 ft msl; I noticed a VFR aircraft tracking northbound towards the final at 5300 ft and climbing. I canceled Aircraft X's approach and told them to maintain 6000 and join the localizer. As Aircraft X descended through 6500 I realized the VFR aircraft was climbing through 5500 so I told Aircraft X to level as soon as possible. I knew he was going to continue to sink since he was descending to 6000 and at that point I wanted him to stop descending ASAP. I issued a traffic alert and turned Aircraft X to the right to avoid the traffic head on and climbing through 5800. I expected the VFR traffic to see and avoid and to turn right as well since that is the rule when head on with traffic but instead the VFR traffic turned LEFT directly at Aircraft X and continued climbing into him. Aircraft X never reported a TCAS RA but I saw them climb out of 6300 (which they had leveled at) back up to 6500. They barely missed each other and the VFR traffic continued northbound. SAT controllers have been begging for an airspace change for a decade now to stop VFR traffic from becoming safety events and near misses within 15 mile final of a major airport and it feels like no one is listening. We need to change the airspace so that VFR aircraft like the one today can't just freely fly through the SAT final without talking to anyone. If that Aircraft X pilot had done anything a second later than he did this would be a very different report. That is the closest I've ever come to witnessing a mid air in my career and it was horrifying.