So most of the time, GPS altitude will closely resemble pressure altitude
You don't mean
pressure altitude, at least not using the conventional definitions, which is flight level (indicated alt with 1013 set). You mean
indicated barometric altitude.
Nice analysis, BTW. It's never been clear to me if aviation GPS receivers use a Geoid model to convert height above the standard ellipsoid. It doesn't seem hard to do. Yet geoid height gets the blame for a lot of the vertical errors.