It may be the guns' official designations, but outside of a one-off trivia blurb, the girls themselves are never called by them in the anime, and neither do official materials (ex. post #1162654, post #1162235). If anything the aliases should be the other way around (with exception to FNC_(upotte!!)).
The series is rather inconsistent with displaying names, but in Episode 1 (chapter 1) Instructor Garand (Garland) refers to L as L85A1 when she shows up late to the firing range (ex: animemanga ). In episode 4-6 (chapters 11-18) during the Jungle team battle all the characters are wearing name tags, which has their full names written out. Ex: scanslated manga and screen cap 1, screen cap 2, screen cap 3, screen cap 4, and screen cap 5
post #1162235 if you look, FNC is referred to as "Funko" which is her nickname (notably one she does not care for), which would give weight that all the other 3 listed are also "nicknames" or shortened names, given that below is their full designation.
The character MP5A2 (younger sister of G3A3 and older sister of MP5K) introduces herself in episode 3 (chapter 19) as MP5A2, but asks that they just call her "MP." The Japanese wikipedia (and ANN) just lists her name as "MP" and lists her full name in brackets like the rest of the cast, which would indicate that the Japanese names listed are only their nicknames.
Fair enough. My argument is for the most frequently used names, but the amount of evidence for their designations (their "full name") is certainly large. I will admit that the tag changes I made stem from English Wikipedia and TV Tropes using the short names rather than the full designations.