Answer

I assume that, halakhically, one can discuss the question of the liturgy used there (if they have a different liturgy) and mixed seating of women and men.
As for the liturgy, I am not familiar with the reality, and every change in the liturgy has to be discussed on its own merits. As for mixed seating, in my opinion there is no halakhic prohibition (in strict Jewish law) against mixed seating in a synagogue. It is an ancient and accepted custom, but in my opinion no more than that. However, if the women expose parts of the body that are considered modestly covered, there may be a prohibition against reciting sacred matters and the Shema in front of them. And even about this there is room for discussion; at least regarding a woman's hair, several halakhic decisors have already written that nowadays, when women are accustomed to leaving their hair uncovered, the halakhic situation is different.
Needless to say, I do not think that what troubles those who would forbid it is any halakhic question whatsoever.