Actually, while for all practical matters it is indeed forbidden for a Catholic to become a Mason, there is nothing prohibiting a Mason from becoming a Catholic
And from a historical note, the Illuminati, an italian body, were formed
by freemasons with rather radical political ideologs, not the other way around. Many spurious organizations utilize freemasonry as a type of applicant pool, as Master Masons (those of the 3rd Degree, or “level” as you called it) have already sworn themselves to various oaths of secrecy, therefore “proving” them- selves capable of keeping secret knowledge, if they show to be good masons over time.
However, it would really be a mistake to suggest that freemasonry has any authority over such bodies, or that they control some imagined world freemasonry governing body, or that such spurious bodies have any jurisdiction over the masonic lodges.
Of course, when such spurious bodies utilize such gradiose titles as “Supreme Lodge of the World” and other such nonsense, it is easy to see why non-masons would be confused. But then, as an American freemason, I could never recognize some body of “illumined Masters” who control all of freemasonry from a jurisprudence standpoint, as not even America has seen fit to creat a single Grand Lodge to which all of the State Grand Lodges would recognize, let alone an international body. Even our “international” sprious organizations are split into multiple bodies that do not recognize the superiority of the others over them, the AASR for example (Northern and Southern).
But then, European masonry is a bit different than ours, as I am told. Ironically, the French freemasons, who were far more successful in their doings regarding political revolution than the more famous Illuminati, are not even entirely “recognized” as legitimate freemasons by much of the world due to their peculiar innovations (it is much too complex to get into here in any real detail).
And finally, as an ex-mo myself, I can also add my two cents that neither are the JW and LDS the same, but also that they are NOT being run by the Illuminati, or even the Freemasons. A little research into LDS history will easily reveal the conflict between them and Masonry. IMHO, the only reason that the LDS church has not banned Masonry from its membership, as has the RCC, is that it is too well known that the majority of the beginning LDS leadership were masons

I think it very safe to say, however, that such is not the case now, and hasn’t been since the “exodus” to Utah (too many under BY’s rule felt that the Masons were responsible for the Carthage incident).