Characterising invisible equivalences by confluent rewrite rules
Yes.  With M=(λx.x x) per the question, consider the rewrite &zeta; that takes M M p to M M.

It is confluent, and so characterises a reduction system over the lambda calculus.  Argument sketch for confluence: since M M is closed, we only need to consider critical pairs of the form M M N1 ... Nk.  These can be resolved.

It is an invisible equivalence, because terms of the form M M I ... I (with zero or more I s) are closed unsolvable terms that reduce only to themselves in the base lambda calculus, thus are distinct, and so the infinite set of these terms is nontrivial, and is obviously equated by &zeta;.

I don't like accepting my answer to my question, so I'll accept an answer from someone who provides a less absurdly incomplete confluence argument.