Assuming the women are dressed modestly and there is no issue here of praying in the presence of nakedness, etc.

But if that’s the custom, then with regard to lower-level singing you also hold that it’s forbidden. Am I understanding correctly?

“Forbidden” is too strong a word.

Is there any other halakhic decisor who permits it? Because the Talmud in Sukkah implies that separation is necessary, and that’s also how Igrot Moshe ruled, among others.

Where exactly does the Talmud in Sukkah imply that it’s forbidden?
Could you post a quotation?

The halakhic decisors here are captive to the battle against Reform and to custom. In my view, this has no real halakhic basis whatsoever (aside from custom, of course).
You probably mean to base it on “They made a great enactment there” (Sukkah 51a–b), as is common in our circles, but that is, of course, an unfunny scholarly joke. If anything, it proves the opposite. Before the enactment they mingled—so you see that by strict law it was permitted. And the enactment, a rabbinic ordinance, was made only for the Temple, at the Celebration of the Water-Drawing. And there is logic to it, since there it was a very great celebration, and there was more concern about the mingling of women and men. That is not the case with prayer in a synagogue. Especially since in the Temple the place itself also matters: because of its greater holiness, one must be more careful there about mingling and revelry, and one cannot necessarily infer from there to other places with a lesser degree of holiness (like wedding halls, for example, and even synagogues).
And this is what Maimonides writes (Shofar and Sukkah 8:12):
Even though there is a commandment to rejoice on all the festivals, on the festival of Sukkot there was an extra measure of rejoicing in the Temple, as it says, “And you shall rejoice before the Lord your God seven days” (Leviticus 23). And how would they do it? On the eve of the first festival day they would prepare in the Temple a place for the women above and for the men below, so that they should not mix with one another. And they would begin rejoicing from the conclusion of the first festival day, and so on each and every day of the intermediate days of the festival, after they had offered the afternoon continual offering, they would rejoice for the rest of the day and throughout the night.