Astronomers discover largest solar system
bbc.com
bbc.com
One more option: "exosolar system".
First, I would call it the Solar System. (Not any solar system, mind you.) That would be in line with existing nomenclature in which the Solar System¹ is the name of our planetary system².
Second, "moon" is not the best term (although I know it's in use like a lot of other "liberal English" words) and I would prefer "(natural) satellite"³ for this generic meaning and leave "moon" for the (Earth's) Moon⁴.
¹ https://en.wikipedia.org/wiki/Solar_System
² https://en.wikipedia.org/wiki/Planetary_system
Full Definition of solar system : the sun together with the group of celestial bodies that are held by its attraction and revolve around it; also : a similar system centered on another star
Wouldn't object on the higher end of this range be regarded as a brown dwarf, making this system a binary star rather than a planet system?
Perhaps if it doesn't make it as a star, then it's not binary? It's not quite a planet system, but maybe one can call it a "Solar system"?
[1] https://en.wikipedia.org/wiki/Brown_dwarf
This might be interesting. In the same article, they call a pair of brown dwarves a "a binary system" rather than a binary star.