A TRACON Controller reported a departure could not comply with the minimum climb rate on the SID for terrain so they gave them a heading off course which placed the aircraft into a different area below the Minimum Vectoring Altitude.

Narrative

I was working Approach with a few VFR arrivals and 1 IFR ZZZZ departure Aircraft X. Aircraft X was filed on a routing that required an MEA of 11;700 ft. Aircraft X was filed on the ZZZZ departure SID which requires a 235 ft per 1NM climb rate through 6;700 ft. They were not maintaining this climb rate which caused them to be approaching an even higher MVA of 8;600 ft. than the 7;600 ft. that they were currently in. I instructed the pilot to expect box vectors for a climb since they needed to be at 12;000 ft. by ZZZZZ; in less than 15 miles. I turned them right to a 270 heading and then realized they were at 7;100 ft.; not the MVA of 7;600 ft. I asked if the pilot could maintain their own terrain and obstruction through 7;600 ft. in which they were unsure. I instructed them to have a minimum delay reaching 7;600 ft. in which case they did so within about a mile of flying. With slow performing aircraft make sure I double check the altitude they are currently at before vectoring them. This aircraft was 15 miles off of the departure end of the runway and had only gained 1600ft of altitude. I had an expectation bias of their current position that they were above the MVA of 7;600 ft.