Is it permissible to take (limited) participation in salat, in solidarity with our Muslim brothers?

I have a Muslim friend who often invites me to come to his masjid after my Friday night synagogue services. While there, I will sometimes pray alongside the Muslims when they take salat. I have read the translations of their service from Arabic into English and find that most of the things said during this prayer are perfectly compatible with our faith. I will repeat the “ameen” during these compatible portions. There are one or two portions however, that mention foreign concepts such as praying for defense against their “satan” figure, or praising Muhammad and his family. When these passages are spoken, I do not repeat the “ameen” but will instead take myself from out of the line of worshippers and recite a portion of the evening amidah or the aleinu.

I genuflect alongside them upon their repetition of “Allahu Akhbar”, as it carries the same notion and intentionality as the phrase “Baruch Hashem”. Considering that Mecca and Jerusalem are in the same direction from where I am, I feel there is no shame in genuflecting in the direction of the temple upon the recitation of a phrase praising G-d.

I would not participate in Christian services in such a manner, but I felt that doing so with Muslims could be justified on the basis of my reading that the Rambam would often pray in Masjids, and my reading that there is a general rabbinical consensus that it is acceptable to pray in the mosques and masjids of Muslims, as they are staunch monotheist Noahides who shun all forms of idolatry. I believe the Muslims are wrong about many things, but I do feel they ultimately worship the same G-d that we do. I felt that my participation could serve as an extra opportunity to both offer additional prayers and to show solidarity with our monotheistic brothers with whom we have had too much strife and conflict. I make my Judaism apparent to those I am with so as not to deceive anyone, and I have received much appreciation from the members of their community for these attempts at solidarity.

However… recently I have begun to feel spiritually unsure about the kosherness of my participation. I considered that perhaps there is a big difference between “praying in a masjid” and “praying alongside the followers of another faith in a masjid”, and that I am potentially rendering myself less pure for my participation. Perhaps I am blowing it up in my head, but I wanted to put the question to folks here. Should I insist on standing apart throughout their whole service and simply reciting my amidahs? should I stop allowing my friend to invite me to these services altogether?

If there are any actual rabbis here I would prioritize their input. I plan to consult my own rabbi at my earliest opportunity as well.