No no no no @turnedworm Firstly you are NOT being unreasonable.
This is 100% his ego in overdrive. He clearly loves the attention from this woman this has been his diet for a very long time, her besotted with him . He loves even more that he is in this very odd love triangle and two women want him.
The fact that he has been seeing her behind your back would be a real red flag. What else is he not telling you? He didn't tell you because he knows you would not approve so just does it anyway.
This screams a lack of respect for you, a lack of maturity and a lack of boundaries.
Did I read right that she greets him with a kiss on the lips? No way in hell should that be happening. You should not have to point out the boundaries that should be created this should be automatic.
All his actions show is that this "friendship" is more important than how you feel and his relationship with you.
He is still acting as though he is single. When you are in a relationship with someone you have to accept that there is some behaviour that is no longer appropriate and his behaviour falls into that category IMO.
I am sorry to say but I would not be suprised if there isn't more to this and if not at this time then watch this space.
At this point this is at the very least emotional cheating.
It's clear he is not going to stop seeing her and she is going nowhere so the question for yourself is what are you going to do about it?