Technically, aren't planets moons of their star? So if a 'planet' has a moon, wouldn't that be a submoon?
No. The difference between “star” and “not star,” within the context of a solar system, is well defined. (Brown dwarfs muddy the line in general.)
The difference between “planet” and “not planet” is a little more arbitrary, with the line between “moon” and “non-moon satellite” being more arbitrary still.
The all-encompassing term you’re looking for is “satellite.”