TRACON Controller reported aircraft descended below assigned altitude and MVA; resulting in a CFTT event.

Narrative

Aircraft X had been cleared for the RNAV XX at ZZZ and was told to execute published missed approach on the go. The aircraft departs ZZZ on the published missed and makes a request for 3;500 ft. and left or right of course for maneuvers. I asked the Aircraft X how much further southwest they would be flying due to the edge of my airspace only being roughly 10 miles from their current position. The pilot responded with 'we're looking to practice some unusual attitudes and then proceed to ZZZZZ for the localizer approach at ZZZ1'. I responded and told the pilot to 'climb and maintain 3;500; that's approved'. I had a departure and overflight conflict that drew my attention. I had already determined that Aircraft X wasn't an issue and was watching my other 2 aircraft. The low altitude alarm went off and noticed the Aircraft X at 2;400. I issued the low altitude alert and the pilot responded immediately and climbed to 3;500. I had no further issues with the aircraft. The MVA in the area was 2;600. Recommendation: 1. I should have given the pilot a pilot deviation. 2. I assumed the pilot was asking for turns left or right; not climbing and descending. I should have asked more questions to the pilot to fully understand his request.