Tried this out and still getting the same sinking issue on pushback. It seems like it has something to do with the "freeze" part of GSX that occurs on pushback, considering it only happens when the aircraft is supposed to freeze.
Sure it happens because GSX needs to freeze the airplane, that's why I asked if it happens with other planes because, when the airplane is frozen, it supposed to sit on its static_cg over its contact points so, maybe, there's something unusual with that airplane contact points.
In MSFS, by activating Devevelopers mode, there's a menu where you can visualize lots of things about the airplane, including checking where the contact points, like gears, have been set.