Why should she be allowed to compete with biologically female athletes, be housed with "real" (sorry, sorry, biological) women, be allowed to use the Ladies' restroom?

I'm not sure why you're using that phrasing, if you're starting from the point of acknowledging trans identity. We don't ask why she should be "allowed" to have the same rights and privileges as other women; we ask if there's any reason to prohibit her from being treated like any other women.

For nearly all circumstances, the answer is "no." There's no reason for treating a trans woman differently than any other women - in the workplace, in a restaurant, shopping, anywhere. You've focused in on the small handful of contexts where biology plays somewhat more of a factor than other place. But even then, there's no reason why trans women can't use the ladies room. There's probably no reason for them to not be housed in women's penitentiaries. In both milieu, the negative consequences of sexual violence are almost certainly higher if you forbid trans women from using the female facilities.

Athletics is a slightly more complicated case - but as has been pointed out, the frequency of participation by trans athletes at the levels where this is high enough to matter is so low that the weight of interests probably leans towards not prohibiting them from competing in their identified gender. Yes, there will probably exist at least one situation where a trans athlete actually places in the top tier, and their genetics and biology might end up being a determinative factor. But again, not only is that going to be an extremely rare occurrence, but that sort of thing happens all the time in competitive sport. By the time you get to the podium level, the athletes who are winning aren't there just because they trained harder or wanted it more - it's because they have genetics and biology that are very close to the optimal for their sport. For example, if you're 5'9", someone like Michael Phelps (6'4") has an enormous advantage over you based solely on his physiology - you're never going to be able to beat him in a race, no matter how hard you work or how hard you train.

Given the harms caused by discriminating against trans women, the balance of interests likely leans towards not prohibiting them from competition in women's sports. If (as BHM suggests) that changes dramatically going forward as trans women encounter less discrimination in society, we may need to revisit it. But right now, with such trivial levels of even participation, that seems like a policy in search of a problem.