TRACON Controller reported an aircraft descended below their assigned altitude resulting in flight towards terrain. The aircraft executed a go-around and diverted.

Narrative

Aircraft X was issued extreme precipation near ZZZ moving toward the field; offered alternate airports. Aircraft X wanted to try to land ZZZ anyway and requested to track the RNAV approach but be cleared for the visual. I coordinated the opposite direction runway with the tower. I offered the lowest MVA altitude to Aircraft X and he accepted it then I assigned it. I issued deviations left and right of course; when able direct ZZZZZ - the FAF. Aircraft X was assigned 1500 but indicated 1600 near ZZZZZ. As he turned inbound toward ZZZ; he indicated 1400. There was other coordination occuring and lots of background on guard and lightning static. I then noticed Aircraft X indicating 1100 as I was transferring communication of another a/c. Then Aircraft X indicating approx 005 as I was initiating a low altitude alert due to the 1100 indication. Aircraft X previously proactively asked me for alternate missed due to the flight conditions; so he was already in the turn. He stabilized and decided on an alternate airport.The loss of altitude resulted in me violating the LOA (Letter of Agreement) by not transferring communication with the a/c in time in addition to violating the Class D airspace. I asked the pilot the reason for the loss of nearly 500' and he said wind shear.He admitted to descending to 1200 on his own and then encountered wind shear. He descended to 400 or 500 feet and climbed when I issued the low altitude alert.