Official statement on the vessel's fate (archived for reference)
On the official BattleTech Forum ("Ask the Writers" section, where canonical rulings and clarifications are given) it was asked in this thread what had happened to the Full Moon, as it did not appear on the rosters in the Field Manual: 3145. Former BattleTech Line Developer (and still an official BattleTech writer) Herbert A. Beas II (as HABeas2) replied on 16 March 2014 that the ship was gone and elaborated two days later:
I was just informed: Sold for scrap. the ship was in sorry shape, and the Exiles could no longer maintain it (no functional WarShip-ready yards in the LC), so she got sold off to another Clan, and was subsequently broken down for parts. It happens.