I don't believe ASME means going from round to flat/square as an essential variable filler metal type. If your using solid filler, and you change to powder, or flux cored filler, then your essential variables will change. Reading QW 404.23 should explain it for you. Most every GTAW solid electrode i've seen has a flat area with the electrode classification stamped on it. That's not a cause for needing a new WPS.
As long as your WPS states solid (or whatever it may be), and that's what your running, no new WPS is needed, be it round, square, flat, or triangle electrode.