The term "walty" (or walt) is generally used across the services for people who are "in" but just a bit.. Jack. Prone to fantasies and exaggeration.
I think her lie is stretching that term though tbh.
The police also describe excessively gung-ho types who tend to buy very expensive accessories and get involved in kerfuffles off duty as "job pissed" but medical emergency services have adopted "Ricky Rescue" for the same sort of situation eg paramedics that are SAR volunteers and special constables or similar.
I can't think of a word for outright faking but if I come across one I'll report back.
Wannabe might be a reasonable fit for now though.
I always thought Walt was never served but pretended they did, bloater was did serve but told everyone they were on the balcony for the SAS when actually they were in the catering corps.
Jack actually bloats her dad over and over again as a Para to the extent he has issued his own correction on her wikipedia entry, essentially the Royal Artillery were attached to a Para brigade for some operation and Jack has decided that means she can tell everyone her dad is a Para, I can only imagine his despair, especially as he seems very proud of his service and is still active in a group to do with it now.
I think it does sum her up though, especially the way she tries to co-opt her Dad and Brother's service into her being 'military round the edges' whatever that means. Even that is all sterotypical, I've met people who are ex-forces who are ultra organised and I've met others who are complete slobs, just like any other walk of life. Jack is definitely the type to do one day of basic training then drop out and wear a veteran's pin without shame.
I think the closest term I can think of for her is catfish, except not just for romantic partners but everything. Like some kind of personality vampire who absorbs whatever she thinks is most advantageous at that point in time until she feels a better one has come along.