I understand the mathematics involved in Fourier analysis, but that is the mathematics -- is the electron ACTUALLY doing that, or was that simply a mathematical/logical proof that correlates highly with what we see?
ISTM your question is meaningless. The best we have to offer on what the electron is ACTUALLY doing is with mathematics that correlates highly with what we see. I don't know what it means for there to be an actuality beyond that.
Even your question/remarks on the "correct conceptual framework" seems to miss the mark. The best we have there is the simplest mathematics that correlates highly with what we see.
All of this mathematical physics has its root in formulas that
were derived based on data collected in labs,
Actually, a very big part of the theory is predicting new and unexpected results that have not been seen in the lab yet. Another big part is when the same mathematics can describe different phenomenon that were previously thought to be unrelated. Lee Smolin provides an excellent description of how this all works in his book The Trouble with Physics. I highly recommend it.