What are the Martin's Maximum consequences of Namba forcing?

Magidor's proof that MM implies that there are no good scales of length lambda^+ for lambda>cf(lambda)=w, utilizes a Namba-style forcing.

I think that I may have found a suitable candidate; namely, the result of Konig and Yoshinobu that $MM$ implies that there are no $\omega_{1}$-regressive $\omega_{2}$-Kurepa trees. The proof seems to have the same relatively direct flavor as those in Baumgartner's $PFA$ article.

Justin Moore asked a similar question a while ago, and I pointed him to my papers with Claverie and Doebler. Even though it doesn't exactly answer your question: Namba-like forcings (in the sense that they are stationary set preserving and make $\omega_2$ $\omega$-cofinal) have many applications in the presence of ${\sf MM}$. The most recent one is my proof with David Asperó that ${\sf MM}^{++}$ implies Woodin's $(*)$, see https://ivv5hpp.uni-muenster.de/u/rds/MM_implies_star.pdf .