ZLC Controller reported failure to correctly issue low altitude alert to traffic experiencing icing resulted in a CFTT event.

Narrative

Aircraft X was inbound for 46U at 160 when the encountered lgt-mod icing. They requested lower and i promptly decended them to the lowest altitude of 130 in a 129MIA. Once there the aircraft had difficulty holding 130 and descended to as low as about 125 in the 129MIA. Due to the proximity to the 46U the MSAW did not activate and initially I thought the aircraft was still above the MIA. This was my second day training on R08 and was not overly familiar with the exact airport location as well as the MIAs. When I checked the MIAs and saw the aircraft was below the MIA I notified them they were low and that the MIA was 129 but did not initially say 'Low Altitude Alert.' After prompting from my trainer that I should do so; I state the low altitude alert phraseology. I also at some point did issue a vector to start to get the aircraft to lower terrain so they could continue their descent but this may have occurred below the MIA. A procedure that would allow a descent below the MIA if the aircraft in a potentially critical situation similar to this and is on a vector to lower terrain and a certain distance from lower terrain could be a useful tool. With today's weather; the vector to lower terrain put the aircraft into a strong headwind that delayed my ability to get them lower to more favorable weather conditions.