A TRACON Controller reported a NMAC between an aircraft on an approach and an unidentified VFR aircraft flying through the final approach course.

Narrative

Aircraft X descending via BRACKEN Arrival into RND. Two airports are just outside of SAT Class C airspace and are on the final approach to RND. This is an area of constant traffic alerts and TCAS events due to non participating aircraft climbing into ILS traffic landing RND. This unknown Aircraft Y was a primary only on RADAR; on the localizer path to Runway 15L at approximately 3;000 feet which is the same altitude for the ILS approach in that area. Traffic was issued 2 times approximately 7 miles apart and then again about 3 miles apart. Aircraft X reported the aircraft in sight and advised it appeared to be Aircraft Y approximately 200 feet below him. At this time Aircraft X was at 3100 ft. or 3200 ft. but should have been at 3000 ft. but due to traffic was not at assigned altitude. Suggestion: San Antonio needs a larger airspace to encompass the very closely spaced and congested airspace around San Antonio and RND which by flight operations is the busiest air force base in the US. Dodging unknown VFR traffic while trying to run simultaneous ILS approaches is not safe; especially in the case where they don't even have an operating transponder and are operating on the localizer at the altitude for the approach. Final approach courses should be protected from non participating aircraft.