A TRACON Controller reported a departing air carrier responded to a TCAS RA for an unidentified VFR aircraft which caused the air carrier to fly below the minimum vectoring altitude.

Narrative

ZZZ Tower requested release for Aircraft X; for a ZZZZZ departure heading east-southeast and turning to East through a pass in the mountains. At the time; I noted two 1200 squawks aircraft north/northeast of ZZZ VOR; approximately 30 miles East of ZZZ; westbound at 10500 ft. and 11000 ft. I determined there was enough time for the release window of three minutes on the ZZZZZ and released Aircraft X with no restrictions. This is a fairly common scenario at the ZZZ sector as the VFR inbounds typically call ZZZ on the north side of the pass while the SID takes the aircraft through the southern portion. At XA18 both primary aircraft that were observed were lost on radar over ZZZ VOR approximately 20 East of ZZZ. Due to mountainous terrain this is also expected. ZZZ [sector] was slightly late on getting Aircraft X airborne and had departed a VFR eastbound North of the ZZZZZ prior to clearing Aircraft X. This too is often typical of the operation in this sector. Aircraft X called airborne passing 6000 ft. on the ZZZZZ; the aircraft was radar identified and told about the last known positions of the VFR aircraft eastbound at 10500 ft. and 11000 ft. the VFR aircraft that departed prior to Aircraft X was still tracking East slightly southeast bound and was climbing roughly the same as Aircraft X and 1000 ft. higher indicated. I was receiving intermittent radar returns from VFR aircraft well North in the pass often times indicating 10000 ft. or higher westbound. I suspected these were the previously observed aircraft that were over ZZZ1 and were no conflict for the SID.Aircraft X was issued traffic on the previous VFR departure that was tracking eastbound 3 miles North of the SID at 9300 ft. almost 1500 ft. above Aircraft X altitude. After reviewing the replay; within ten seconds after the traffic advisory was given the VFR target turned to a more southerly track; but given the speed and rate of climb Aircraft X was going to pass a mile behind the aircraft. I issued a control instruction to another aircraft and as I issued this instruction a 1200 code acquired 1 mile west of ZZZZZ1; a fix on the SID; westbound at 10000 ft. This put the departed VFR aircraft and the newly acquired 1200 in an unsafe position for each other. The newly acquired 1200 code then began descending to avoid the previously departed VFR aircraft. I issued a stop to the climb of Aircraft X at 9000 ft. while the aircraft was at 8600 ft. the pilot responded and said they just got the resolution advisory. I issued traffic on the new 1200 code but due to overlay I issued the wrong altitude of 11000 ft. of the departed VFR aircraft not the 9800 ft. of the newly acquired aircraft. I observed Aircraft X at 8200 ft. as they stated the traffic had passed and climbed the aircraft to 16000 ft. They were at a portion on the SID that goes through an MVA of 9000 ft; but on the SID the previous fix was at or above 8000 ft. and the next was 11000 ft. Due to how fast this all transpired I issued the climb but did not give the low altitude alert. Later the previously departed aircraft called to pick up an IFR flight plan on file from ZZZ southeast bound. Due to the nature of terrain; unfortunately our ZZZZZ SID shares close airspace with VFR inbounds from the East daily. Changing the SID on an East flow for aircraft eastbound would be a cumbersome process; but as the SID is currently published there is often IFR departures with non radar VFR aircraft within 5 miles of each other on a daily basis. There could possibly be a rerouted SID utilizing the initial portion of the ZZZZZ2 SID that climbs southbound then turns to the southeast; this could deconflict up to 50% of the ZZZZZ departures from the pass; but would result in possibly a change in climb gradient for the pilots. ZZZ already states the allotted time for releases; and I should have canceled the release and utilized the DVA when they missed their time. Also there could be a possible addition to the Letter of Agreement (LOA) with ZZZ that VFR release of IFR aircraft should be coordinated to deconflict with other departing traffic and we can ensure safe proximity from terrain when they are requesting their IFR pickup.