Thank you!
I was genuinely surprised not to be able to find a Capellan unit that used one in TRO 3050. The Thunder in TRO 3055 is stated as being the first post-Clan Invasion heavy mech and, despite running the AC/20, doesn't have a Gauss rifle config. By 3055, even Wolf's Dragoons were manufacturing and employing them (Grizzard on the War Dog).
So that gives us:-
The Demolisher variant in 3052.
The first use with a named weapon is, I think, the Regulator which comes in by 3057 with the Inokuma Gauss Rifle.
I think the first Battlemech is the JN-G8A Jinggau with the Zhi-tong yao in early 3059.
In the Aerospace sphere, even the Transit doesn't get a Gauss Rifle variant to address its range issues.
Given the Gauss rifle's general good performance and relative ubiquity, it seems an interesting quirk of national production.