Just to unpick this, for a minute. Are you saying that in a lesbian relationship there could be an abusive partner? Yes, of course there could. Is there much chance that if the abused woman went to a refuge that her partner would be allowed in? Not really. The refuge would have details of the partner, probably a photo, and would block her access.
The point about refuges is not that abusive partners are likely to follow their victims, but that they are places where vulnerable women and their children have gone for protection. If a man who wanted to target vulnerable women wanted to, he could declare himself a woman and be admitted. There are huge issues for refuges with this, as if they refuse access to men who say they are women they are accused of transphobia. The end result is that women who need space away from men, who are potentially traumatised, find themselves housed with male bodied people, some of whom are of the obviously masculine variety.
I dare say that a psychopathic woman may pretend to have been abused in order to gain access to a refuge, but respectfully suggest that this is more likely in the plot of a B movie than likely to happen in reality. The vast majority of violent criminals are male, and the real issue is, as ever, that men are (generally) stronger and more powerful than (most) women.
This strength differential (which, before you bring in your friends who confound this stereotype, is not universal but usual) is the reason for having safe spaces in the first place, and the reason that they are based on sex, and not on gender. It is not that posters on this thread are too uninformed or dim to realise that not all women are good or behave well. It is that we have the sense to understand that sex-based safe spaces are there for a reason.