It's not. As things I've posted earlier in the thread demonstrate, and people keep conveniently ignoring: women's shelters - at least in the UK - have been using self-identification for years. The whole question of "would it be a problem if" misses out on the fact that it's not a hypothetical. It's what actually already happens.
It doesn't seem to be a significant problem.
I think this is a misrepresentation of the actual facts.
I am fairly confident that, back in the day, and that day was no more than 10 years ago, maybe 15, many shelters in the UK would turn away a male bodied individual. Of course, in order to do so, they would have to know that the person was male bodied, so it was kind of a "Don't ask. Don't tell." situation.
More recently, I'm sure that shelter staffs were more accepting and more likely to accept a transwoman, but under the implicit or explicit instruction to maintain some sort of plausible attempt at passing.
And most people would be just fine leaving it to the discretion of the staff that way. I don't live in the UK, but going by what has been said in these threads the shelters are expected to accept anyone self declaring as a woman, regardless of whether they obviously are not, and regardless of whether they are making the least amount of effort to pass. Based on anecdotes in this thread, it would even be the case that some people are rather blatantly proud of being male bodied and want people around them to know about it. That behavior is consistent with some that I have seen in trans-friendly situations. The trans person does not want to actually pass as female. They go out of their way to make sure that people know that they are males who present as females.
And....whatever. The trans-friendly place where I encountered this behavior was a game convention. I would prefer that the transwoman not make it so obvious what was under her skirt, but as long as she doesn't actually show it off, and she rolls the dice when it's her turn, fine. Enjoy. However, in a women's shelter, that kind of behavior wouldn't be cool, and I think the staff should have discretion to eject those people.