ABQ Local Controller reported a departing aircraft was issued a traffic avoidance turn and altitude restriction; which resulted in entering a higher MVA.

Narrative

Aircraft X requested [Runway] 3 for departure. Volume wise we determined it better for [Runway] 8; so Ground taxied aircraft to [Runway] 8. Aircraft Y was orbited at 105 5NNW of the field. Aircraft Z; a C172; was inbound from the southwest for [Runway] 8. I saw Aircraft X approaching E1 and squared off Aircraft Z to give me proper wake turbulence. I reached out of Aircraft X and told him Runway 8 LUAW (Line Up And Wait). I did this twice and no response. When I turned to Ground to have him try Aircraft X; he was talking to the aircraft and told him that he was good right there on E1. I intended to reach out and cancel the LUAW and just let the C172 land. At that moment; I observed Aircraft X continue on E1 and cross the hold lines for 8. I reached out that time and said RWY8 cleared for takeoff via the SID. I was concerned about the wake turbulence. He started the takeoff roll. About a mile off the departure end I remembered Aircraft Y was out there and turned Aircraft X to a 310 and stopped his climb at 90. He was at 89 in an 87 MVA and turning away from higher terrain and MVAs. He did slop into an 110 MVA. Aircraft X took a moment to stop the climb and went to 96 before he said he was going back down to 90. I said Roger.I understand that I cannot turn someone below an MVA outside of the DVA; but Aircraft X was above the MVA when the turn was issued and the turn lead him to a lower MVA that 90 cleared. I understand performance of aircraft and pulling a last minute change on an aircraft will cause them to slop turns. In this case that's what happened. The turn was a sloppy turn.I don't hold the hold bar crossing against the aircraft because he could have heard me and have been cross channeled. I don't hold the altitude above the clearance against him because I did issue the instruction late.I pulled the aircraft off the SID because I felt like I was going to hit Aircraft Y with him. I felt like the safest decision was to turn Aircraft X away from the traffic. I could have allowed him to continue the climb to 100 because Aircraft Y was at 105. Once I saw him established in the turn I could have tried to apreq higher for him. My decisions were made within seconds and I didn't feel like I had time to call radar and to coordinate anything different. I was simply trying to not hit Aircraft Y. Hindsight is 20/20 and looking at it now I feel like I could have safely let the aircraft continue to climb; but I was not talking to Aircraft Y and with the wake turbulence it was a risk I was not willing to take with seconds to fix it.I feel a little discouraged to be told by Management that I should not have vectored an aircraft out of a situation that I thought would be been a pretty dangerous NMAC. I did not feel like Aircraft X altitude in the turn was an issue with the terrain. I watched him out the windows in the turn and felt that the turn would continue to lead him to the lower MVA that 90 cleared. The mountains are closer to the field on the south side; so a south turn would not have helped.The aircraft has not been amended to RV at 9. It slipped through data. It slipped through Ground; and in the moment of trying to not give Aircraft Z wake turbulence; it slipped through me.