I always assumed that the term "flatulence" was "wind" that specifically emanated from the alimentary canal out the anus. Yes, I've been to numerous sleep-away summer camps as a teenager, and I have heard those annoying armpit noises. It's not "flatulence", but, I see that you think that it sounds like it, so, fine, I'll go along with your idea...
I entered the term "flatulence" in Google and it provided another definition:
inflated or pretentious speech or writing; pomposity
So, this definition certainly relates to your claim that armpit flatulence is speaking from the mouth. Moreover, it is pompous speech.
Well, Pirkei Avot 4:4 says:
מאר מאד הוי שפל רוח
This has two definitions that apply to your question:
1 - "One should be exceedingly humble". So if that armpit noise is pompous speech, then, this Mishnah suggest that one should not do it. (It doesn't say that one is prohibited from doing it; it suggests against it.)
2 - "One should make low wind" - I.e., if you're going to make armpit flatulence at all (I thought it was a stupid habit - even as a teenager. It's still "stupid", but I'll admit that kids do stupid things.), at least don't be noisy.
In terms of halacha, BTW, one may not say Shema where there is a bad smell such as one from alimentary flatulence. This armpit noise does not emit any foul odor. It's a different problem if the armpit, itself, smells bad, regardless of the flatulence.