ARTCC Controller reported an aircraft flying in icing and snow conditions was unable to climb above MVA resulting in a CFTT event.

Narrative

Aircraft X came into my sector at 10;000 ft. They were on an IFR flight plan heading to ZZZ1. 10;000 is fairly low for the terrain in the area and routing was going to be needed for Aircraft X to stay at 10;000. Roughly 5 minutes into my airspace Aircraft X reported getting into light rime icing. I advised them of my minimum IFR altitude in the area which was 9;000 but that I could only do that for a short amount of time as their was higher terrain ahead that would require climbing back to 10;000 eventually. At this point Aircraft X changed their destination to ZZZ. I cleared the aircraft to ZZZ and soon after they requested the descent to 9;000 as they reported 'We are starting to get a little slow here'. Aircraft X eventually requested the RNAV [Runway] X approach at ZZZ and I cleared them to the IAF of ZZZZZ. Throughout this communication I advised Aircraft X of an airport south of their position that was located in lower terrain; although that airport did not have an IFR approach and was overcast. I also mentioned possibly turning around away from the area of icing and again where there was lower terrain. Aircraft X reported they were going to continue on course at 9;000 for the approach. It was at this time I advised Aircraft X that I could not legally clear them for any approach at ZZZ at 9;000 and that I would have to have them climb to 10;000 for a clearance. Aircraft X acknowledged this information with 'roger'. As Aircraft X approached my MIA of 10;000 I advised them of a low altitude alert for the MIA and re-iterated I could not clear them for the approach; again receiving a response of 'roger'. At one point Aircraft X reported breaking out of the clouds so I asked if they could turn direct to ZZZ to avoid terrain or climb and they responded ' we are going to continue inbound at 9;000. Once they entered the MIA block I gave another low altitude alert and had the aircraft report established over a fix on the approach that had a published altitude of 9;000 and verified their altitude. Aircraft X eventually reached the fix with an altitude on my radar indicating 8700 ft so I had them verify their altitude which they reported level 9;000. I cleared them for the remainder of the IFR approach; terminated radar services; advised to cancel the IFR on the ground; and sent them to the advisory frequency. All while this was happening I also had multiple IFR approaches to multiple other airports in my airspace; arrival and departure traffic in conflict from ZZZ2 approach; Overflight traffic making requests and general frequency congestion as the sector has 4 transmitters and aircraft cant hear each other due to high terrain. I can't say I would have a recommendation other than the sector should have been split off as I was working a combined configuration with way too much IFR approaches due to the weather of the day. I have a feeling our staffing level for the shift is a factor in this as we only had 6 people for a shift that should have 9 and I don't think the conditions of the day especially for a sector that can get very complex with winter weather was considered. I did have to request a D-side in the middle of this situation.