CS, if you do a search for the yichus of the Maharal you will find that there is a disagreement whether he was descended from David Hamelech. That surprised me a few days ago when I first saw it, but it’s there in black and white (pixels).
Secondly, if the Maharal is indeed a descendant of David Hamelech, it is through Rashi (who had no sons) and the nesi’im in EY who were descended through Shefatya, as per the gemoro in Kesubos 62b, not through Shlomo. All this can be seen from a search.
The search also showed me that it is not only in Peirush Hamishnayos that the Rambam says that Mashiach has to be descended from Shlomo Hamelech. He writes the same in Iggeres Teiman:
ודבר זה אחינו יסד גדול מיסודי אמונת ישראל והוא שאי אפשר שלא יעמד מזרע שלמה איש שיקבץ נפוצותנו ויאסוף חרפתנו וגלותנו ויגלה הדת האמתית וישמיד כל מי שימרה דברו כמו שהבטיחנו הקדוש ברוך הוא בתורתו
And wonder of wonders, I discovered that Rabbeinu Bachya – whom you have quoted regarding the disappearance, for want of a better word, of Mashiach – also writes the same on Devarim 30:15:
ביאת המשיח שהוא מזרע שלמה
Thirdly, the Lubavicher rebbe’s yichus to the Baal Hatanya is via the Baal Hatanya’s daughter, so he is not ben achar ben that way even if what I wrote above about the Maharal is incorrect, and the Maharal is descended from David Hamelech ben achar ben.
Simply put, the Lubavicher rebbe is allegedly (I believe it to be true, but I only know it from Lubavich sources, and you may not be surprised to hear that I don’t find them always 100% reliable) ben achar ben from the Tzemach Tzedek whose patrilineal line is unknown.
One other interesting point that I realized today, Bar Kochba/Koziba’s name was Shimon, so clearly R Akiva did not think that Mashiach’s name has to be Menachem.