Here's the thing: "Invader" was a nickname/colloquial name for the A-36A Mustang.
It originated in the 12th Air Force in the MTO, probably in 1943 (after the 500 A-36As ended production). A pilot in the 86th FBG, whose name I could dig out of a book, but I've forgotten it, after pilots and groundcrew referred to the plane as "Invader" because, as they said (I'm paraphrasing), "we're always invading and taking over enemy airfields, etc." This pilot went so far as to petition the USAAF and NAA to change the name from "Mustang" to "Invader," but that was a non-starter, since that name had already been given, OFFICIALLY, to the Douglas Aircraft Company's A-26 Attack/Light Bomber aircraft.
"That "Invader" advertisement has been seen for a number of years, along with ones that specifically call the A-36 "Mustang." Refer to the bottom of this Allison ad from 1944. The association with the "Mustang" name for the A-36 had appeared in other ads as well.
Let's not forget that today's USAF members refer to the F-16 Fighting Falcon as the "Viper," and the A-10 Thunderbolt II as the "Warthog" (or simply the "Hog").
You have seen my copy of the Dutch Kindelberger-to-Col. Arthur I. Ennis Telegram from 13 July, 1942, correct? It was sent to me (as stated before) by the Boeing Historical Archivist almost 4 years ago, along with the cover of that A-36A Preliminary Illustrated Parts Catalog. I might add that he sent me that without my requesting further information after some initial emails between the two of us.
I would be remiss if I didn't mention that NAA themselves, referred to the A-36 as "Mustang" more than once in their newsletter. Strangely enough, after many hours of digging in the "Flying Magazine" archives in Google Books, I came across a letter to the editor (during the WWII years)
The "Invader" name, I will admit, "made it into" some USAAF documents, mostly write-ups by pilots, etc.
The simple fact remains that "Mustang" is the only OFFICIAL name that the A-36A ever had...
This is from the May, 1944 issue of "Flying Magazine," and ... well this "Letter to the Editor" says it pretty well (the line about "Apache" being chosen by the manufacturer...is a conflation of the the "house name" for the two XP-51 aircraft given to the USAAF from the NA-73 batch (first production Mustangs, the Mustang Mk I in the RAF) built for the RAF, AND later, to the 57 (or 58) NA-91 aircraft held back by the USAAF, from the 3rd batch of Mustangs built for the RAF and called "Mustang Mk IA" in the RAF 9 (these were the first aircraft purchased under the terms of the Lend-Lease Act of 1941, by the USAAF for a nation fighting the Germans, Italians and Japanese before the US was involved in the War. AND, once again, the USAAF never called the A-36A (the only model made, BTW) anything but "Mustang."
I'm glad to answer any more questions, and hopefully a new article in a printed magazine is forthcoming later this year that will help solidify the Mustang name for the A-36A.
Cheers.
Tom