A Tower Controller reported they instructed a departing aircraft to turn below the Minimum Vectoring Altitude to avoid a conflict with an arriving aircraft to a parallel runway which initiated a go around on their own.

Narrative

I was the Local Controller providing services. Aircraft Y checked in #2 on about a 7 mile final. I clear Aircraft Y to land. Shortly after I cancel Aircraft Y landing clearance and tell them 'traffic will be holding in position air carrier'. I then see Aircraft X short of XXL ready to depart. I advise Aircraft Y traffic will not depart and that he is cleared to land XYR. I coordinate with ZZZ [TRACON] that I will be departing the aircraft in a different 'Run Down' order and Aircraft X will be #1. I then clear Aircraft X for take off Runway XXL. As Aircraft X is airborne and passing the Tower; Aircraft Y advises they are going around. My eight years of experience allows me to predict that my two IFR aircraft are on a course for extreme IFR loss in the next 60 seconds. I asked Aircraft X if he could make left closed traffic and remain in the pattern. The pilot complied and my CIC (Controller in Charge) coordinated what was happening with ZZZ [TRACON]. When Aircraft X was in the left downwind I had them maintain visual separation with an air carrier arrival on a 4 mile final for XYR. Aircraft X continued his left pattern while climbing; then once they were reestablished on Runway heading I transferred radio control to ZZZ [TRACON]. I knowingly took my IFR departure off their SID and gave them emergency vectors below the MVA because my experience gave me confidence that Aircraft X was capable of the maneuver and weather conditions were adequate to allow it. I determined this was the safest course of action in the moment and worked towards it without hesitation. ZZZ airport has no 'outs' in the event we have an IFR aircraft go around while an IFR aircraft is departing. This needs to be addressed and a resolution better than breaking MVA; or having a massive loss as aircraft climb out on top of one another. This is a clear and ever-present danger that is completely ignored by the FAA.