Since the question here is what women would prefer, I haven't done an extensive search, but it seems at least a majority of women support allowing people to use the bathroom of the gender they identify as:
https://www.pewresearch.org/religion/2016/09/28/3-public-spl.... Interestingly, men are more strongly opposed than women.
I'll also leave this comment for thought. You're arguing that allowing transgender women into spaces reserved for women (unqualified) degrades the utility of them. I'm actually somewhat sympathetic to this argument, but not universally. I think allowing transgender women into women's sports is maybe not right, because transgender women don't have the same underlying physicality as women who have grown up with female hormones. I think it's an interesting discussion about why we perhaps somewhat arbitrarily segment sports in this way but not others (why no under 6 ft basketball leagues?) and the role that genetics plays in sports vs training and practice, but I think it's a generally effective means of allowing a group (generally women) to have meaningful leagues where it's not just women + men who aren't very good but have a genetic advantage. As well with regards to prison, since this is already a segment of the population (people in prison) who have committed some crime, so trusting them in the way we might trust random people from society to act may not be the right choice.
That said, your comments make it sound like you don't respect the existence of transgender people in general.
In particular, "I don't see how this is a self-contradictory position. Women who've decided to call themselves men are in fact still women.", and "It's a reasonable assumption that most women prefer to share female spaces with other women, regardless of how they look, and not with male intruders who have decided to disregard women's boundaries for their own pleasure while falsely identifying themselves as women.".
The first is I think clearly denying the right of someone to identify as transgender, and the latter is a fallacy. I agree that women would not want to share the bathroom with "male intruders who have decided to disregard women's boundaries for their own pleasure while falsely identifying themselves as women.", but this is not an accurate description of transgender women.
If you think it's an inaccurate description of how you feel (that you do in fact respect transgender people) I would suggest that being more careful in your debate might help you to convey your arguments effectively.
