Tower Controller reported having assigned a departing aircraft an altitude below the minimum vectoring altitude.

Narrative

Aircraft X was on a practice VOR approach to Runway XXL. I departed Aircraft Y off Runway YY when Aircraft X was 5-7 miles from the airport. Approach Control was vectoring Aircraft Z for an overhead pattern to Runway YY.Once Aircraft X reached approximately a 3 mile final to XXL I saw that Aircraft Z was going to overtake Aircraft X in the overhead. I instructed Aircraft X to maintain at or below 7500 ft. on the climb out. As soon as Aircraft Z checked on I reiterated that the overhead pattern altitude is 8000 ft. I had other traffic to Runway XXR so my only option for Aircraft Z was to break to the East (left break). I assigned Aircraft Z the departure end break. I used the departure end because I was concerned about the spacing between Aircraft Y (now East of the airport on a 130 heading) and Aircraft Z.I assigned a 100 heading to Aircraft Y to increase spacing from the overhead. At the same time; I believe Aircraft X was trying to ask me some questions but it was blocked by read backs and by Aircraft Z being on UHF. I reiterated to Aircraft X that I wanted them to remain below 7500 ft. to de-conflict with the overhead at 8000 ft. I exchanged traffic between all the aircraft. I believe I made a mess of the phraseology on the traffic calls because I was very uncomfortable with how the situation was developing. In the end; Aircraft X did remain at or below 7500 ft. and Aircraft Z stayed at 8000 ft. until past Aircraft X. Aircraft Z made their overhead inside of Aircraft Y and landed without incident. They were all very close though. Aircraft Z passed between 500 to 800 ft. above Aircraft X and Aircraft Y was only about 1.5 miles East of Aircraft Z's overhead pattern. I don't believe I lost aircraft separation at any point. I believe I may have assigned Aircraft X an altitude of 'at or below 075' and assigned runway heading while Aircraft X was still at 8000 ft. MVA.The aircraft did not have to descend to reach that altitude and was headed towards lower MVAs. There was no particular reason to assign Aircraft X the runway heading; I could have just given them the altitude restriction. But; I was very uncomfortable and over-controlled that aspect of the situation. I am aware that I can not assign a heading and an altitude restriction below the MVA to VFR aircraft. I felt that I had very few options in this situation. Vectoring an overhead to pass directly over landing traffic usually works well; but it does not work well when the preceding aircraft is making an option back to radar. In this case; I believe the approach should have allowed more room between Aircraft X and Aircraft Z. That said; the aircraft maintained around 400 kts. most of the way down final which is unusual in my experience. I could have alleviated the entire situation by having Aircraft Z orbit at initial; but they were extremely fast and descending rapidly and it just didn't occur to me until right this moment. I was focused on trying to make the situation work rather than stopping what was happening.