It's all about the Hairpin
A more revealing title for the film would have been: The hairpin.
Because that is what this is all about.
It's an ancient chinese artefact from the Geisha Period, and it's a talisman forged with special powers of attraction, and with a spell.
This spell is a very fatal one: it makes it's carrier irresistible sexually to others, both male and female, but it will never let it's carrier have them for a real relationship: it exploits the lust-force in them, leaving them depleted and in denial afterwards.
The answer - therefore - to the question:
why did Catherine sleep with Chloe?
is: because she could not resist her, the carrier of the Hairpin.
Chloe however, as the story unrolls, now first experiences the working of the Hairpin towards females, and hopes this time it will work out as a steady bond between her and Catherine, whom she meets at the public bathroom.
The moment Chloe has this realisation, is when she says to Catherine: you're turned on by this, aren't you ?
The message at the end of the movie, where Chloe has killed herself to escape from the fatal spell, seeing it doesn't really create a bond with her in Catherine, other than sexual attraction,
is that the Hairpin once again changes from it's owner to the next,
and now Catherine will experience the same fate as Chloe before, who was also estranged from, and disappointed in, relationship with a man she loved.
The Hairpin does not make you fall in love with people, certainly not: it's a Geisha's magical seduction-instrument, and falling in love with her clients is not acceptable for a Geisha.
It however gives you the greatest seduction powers, because it makes you sense what people need, even if they don't know it themselves.
Chloe used this power on David, because Catherine let her (which surprised her: "people like you, being let into my life").
But David also banishes Chloe out of his mind, he doesn't fall in love with her - he has even completely forgotten about her being with him. It is all the power of the Hairpin.
Every attempt to analyse this movie from logical or known human behaviour of the players will fail, because of the magic factor in it, that drives them, sometimes against all logic or free will.
It is the story of an ancient tool, from a past era, where Geisha Power was playing a very essential role in human society. And therefore becoming absolute.
A power that now, passed on to the mother of Chloe, as descendant of a line of Geisha's, creates havoc in modern relationships and drives it's carrier to utter despair for real relationships,
which it was never designed for to create.
Relationships also, which sexual attraction was never designed for, to create.
Sexual attraction being designed to ensure procreation. Not relationships - they are created by the heart.
Maybe there's a message there, that transcends the plot.