I think it probably is the "A," but I also don't see any good reason for her to be referencing her role in Avalanche on the shirt she wears to her day job as an Applebee's hostess.
I was so happy to have concluded that probably-Nellie does NOT advertise her affiliation on her name tag, using a font specifically associated with Avalanche. It seems ill-advised at best, unless she hopes that people like me read it as a K instead of an A.
To be fair, Avalanche is a bit more fully established as an organization in
Remake as compared to the original game, which is something we've known ever since we got the super HD renders of the characters and
could read the text on Barret's dog tags. Also, as a quartermaster her main role is as a "master of quarters" being lodging and troop accommodations and not really involved in anything where she'd typically be interacting with non-Avalanche members while in uniform.
More specifically in the US sense, since Japan has more exposure to the American military,
the Quartermaster Corps function in peace and wartime and take care of all sorts of basic sustainability supplies and efforts to ensure that the army is well-maintained in case they need to mobilize. They mainly manage things like food, fuel, material maintenance, etc. and aren't involved with weapons, medical supplies, or other things like that (although she'd have been who they got the parachutes from). Essentially Nellie, Al, & Finn all have very supplemental roles and enabling roles to Avalanche that don't deviate all that far outside of what people would do in normal jobs around the Slums – all of which are also designed to keep them out of the way of being seen by anyone else. It's WAY far off from what Barret, Jessie, Biggs, Wedge, Tifa, & Cloud have been getting up to.
Being a supervisor over a barracks' supplies & provisions, matches what it looks like when we see the scene of everybody meeting up. Having some sort of official attire means that even if there are troops from other cells passing into their location (like Yuffie & Sonon), she's clearly called out as the authority over what takes place there, so it makes functional sense for her role in costume design.
And when it comes to subtlety, I think that a little patch like that still pales in comparison to Barret's "I have a literal machine gun for an arm" as a visual identifier.
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