[Bêthberry]

Given that Freemasonry is a private organisation devoted to secrecy, arcane rituals, and hierarchical progression through steps or ranks, that does not sound like something I think Tolkien would be deeply interested in.

Secrecy, especially a sort of conspiratorial secrecy (there's lots of freemasonry themes in conspiracy fiction, going back to the 18C) is something far from the values which Tolkien espouses positively in his tales. I can't see this being attractive to him.

Serious rituals of profound meaning Tolkien participated in throughout his life, more so in his later years. I can't see him needing the sanctification of the little apron with bells.

His professional life was rife with petty progress through the ranks of the academic hierarchy. Why would he need to engage in something similar in his private life or leisure?

Male bonding and comaraderie Tolkien found through the Inklings. Why would he need another boys' club?

I can't recall either that he was much interested in the new findings of Egyptology.

Of course, I could be wrong about him on all counts. He did actively pursue the profit of American money and there's lots of freemason symbols on it. (Most of the signees of the Declaration of Independence were freemasons, iirc.)