A Center Controller reported an aircraft could not maintain assigned altitude due to weather conditions and descended below the Minimum IFR Altitude.

Narrative

I was working the aircraft as usual; he had a block altitude of 10000 to 11000 ft. from the previous controller I believed because of downdrafts. I eventually gave him pilot discretion descent to my lowest altitude in that area of 9000 ft. and he slowly descended. He had ATIS X-ray and was planning the Visual Approach. Pilot requested lower and I told him I didn't have lower due to terrain in the area but offered to vector him a few miles to the west where I could get him 400 ft. lower. He said he wanted to stay right there and maintain 9000 ft. which he did until I was almost finished giving a position relief briefing to the next controller. The pilot said he could not maintain altitude. I plugged into the empty sector next to that sector and monitored the frequency and coordinated with ZZZ Tower while the new controller gave the low altitude alert and vectored the airplane towards the VOR/DME XX approach so he could safely get lower and land. Tower told us during this that the pilot was unlikely to get the visual on the field even after being called in for the visual approach 25 miles north of the field. So the pilot was asked fuel and souls on board and whether he could do the VOR/DME approach. I pulled up the approach plate and started a track on a fix he was near and was measuring the aircraft position to try to determine where the worst terrain was and how best to avoid it. The airplane dropped to 7500 feet in the area where 7402 feet was the highest terrain but he was past that already as best as we could tell. In the end; he joined the approach and was cleared and landed safely at 55 past the hour. We told tower and management that we were [requesting priority] for the pilot in order to vector as necessary below the MIA. Pilot was in conditions he shouldn't have taken that airplane into. Pilot also did not convey the seriousness of his situation and was unwilling to be vectored for a lower altitude and was unable to climb. Pilot stayed calm but I knew that he needed to be watched; I just didn't know what else to do without declaring emergency and vectoring the Pilot in Command against his wishes before he started losing altitude. I think we did everything we could have in that situation with the information we had. Perhaps communications between controller and pilot could have been clearer.