SAT TRACON Controller reported an airborne conflict; a MVA violation; and airspace related problems.

Narrative

Aircraft X was on BKN and Aircraft Y departed VFR out of 1T8 and this pilot was not talking to anyone. Once Aircraft X turned towards the localizer; I issued traffic for a climbing VFR target approximately 3NM away. Aircraft Y then called and asked for VFR flight following and on the scope; they were at 027 NE bound. I immediately gave him Aircraft X traffic and altitude. Aircraft Y continue to climb into the issues traffic. I told Aircraft X to climb to 040 immediately and they read back; 'Say again'. At that point I keyed up and just said CLIMB. Aircraft X descended instead to 025 which is below the MVA. I was given a record of conference by my supervisor and OM because I didn't give a traffic alert or a low altitude alert. In my defense; Aircraft X was going 340kts and after issuing traffic to both aircraft; I felt that I didn't have time to issue the whole proper phraseology for a traffic alert. I felt saying climb to 040 immediately was quicker and to the point. As far as the low altitude alert; after the fact; I wasn't even thinking about that. I was just glad two aircraft didn't collide. SAT continuously has an issue with either 1T7 and 1T8 underneath the RND final and 5C1 underneath the SAT final. This seems to be a daily occurrence where the VFR aircraft or gliders feel they are entitled to climb up into a very active final approach course. We need to have something in place where these VFR pilots are staying below 025 until some point away from the final and the downwind. I know it may be inconvenient for them; but it is extremely unsafe with what they are currently doing.