I have always assumed that when the government in Bavaria (Germany) outlawed membership in secret societies, and especially the Illuminati, that the Illuminati just went underground, and that is when they infiltrated the Freemasons.

I came across this book, written in 1798 (just 23 years after the original founding of the Illuminati) by John Robison. The official title is:

"Proofs of a Conspiracy Against All the Religions and Governments of Europe, Carried On in the Secret Meetings of Free Masons, Illuminati, and Reading Societies, Collected From Good Authorities"

That's quite a title. I have not read the entire thing, but looks like it could provide some interesting insight by someone who lived it at the time.

In the intro, Robison says:

I have observed [the] ... diffusing and mixing with all the different systems of Free Masonry; till, at last, AN ASSOCIATION HAS BEEN FORMED for the express purpose of ROOTING OUT ALL THE RELIGIOUS ESTABLISHMENTS, AND OVERTURNING ALL THE EXISTING GOVERNMENTS OF EUROPE. I have seen this Association exerting itself zealously and systematically, till it has become almost irresistible: And I have seen that the most active leaders in the French Revolution were members of this Association ...

And, lastly, I have seen that this Association still exists, still works in secret, and that not only several appearances among ourselves show that its emissaries are endeavoring to propagate their detestable doctrines among us, but that the Association has Lodges in Britain corresponding with the mother Lodge at Munich ever since 1784.

... I can show that this was all a cheat, and that the Leaders of this Association disbelieved every word that they uttered, and every doctrine that they taught; and that their real intention was to abolish all religion, overturn every government, and make the world a general plunder and a wreck ...

It's a free pdf you can open and read. Worth a look if you are interested in the subject.

https://www.conspiracyarchive.com/PROOFS_OF_A_CONSPIRACY_John_Robison.pdf