I found out that my boyfriend has a fiancé and hasn’t broken up with her

As the title says, I found out about two weeks ago that my boyfriend has a fiance. He was showing me some text messages on his phone and I saw a text that said “on my way home baby,” sent to another woman. I grabbed his phone and tried to read more, but he basically wrenched it out of my hands. When I questioned him about it, he vehemently denied anything and promptly deleted the message thread, claiming that I “didn’t need to read something that happened before we got involved,” (we’ve been together for about 5 months now) even though the message was only a couple down in his message thread and obviously very recent.
I’ve been cheated on before, so this wasn’t my first rodeo; I remembered the woman’s name on his phone and checked her out on Facebook – that’s where I found pictures of them together. I confronted him about it the next day and there was more denial until I told him that I had found her Facebook and had seen the pictures. After some arguing back and forth, he eventually admitted that they’ve been together for about 3 and a half years and were also engaged. Great. Real stellar.
Since I’ve found out, we have stayed together, but things are obviously very strained. He repeatedly tells me that I’m the one he loves and that I’m the one he wants to be with and that he doesn’t want to be with her. He told me that he loves her, but isn’t in love with her and that he has a more of a “caretaker”-style relationship with her. She periodically lives with him, but they don’t sleep in the same bed (this was something that his mom actually confirmed to me when her and I ended up speaking about this) and that they haven’t had sex since him and I became involved. Since I’ve found out, he has also shown me messages from her (but not all of them, I know); she obviously cares very much about him and has said things like she moved out because of his actions – his lack of being around (because he was with me), etc.
Anyway… Be that as it may… I’m torn because he hasn’t yet told her. I don’t doubt that it would be a difficult conversation to have and I’m not unmindful of that; however, I’d naturally assume that if he truly wanted to be with me, this conversation with her would have already taken place. There is a lot happening at work right now (oh, we’re coworkers, btw – in positions that require our relationship to remain a secret or face termination, so that’s a lovely and added complication to this whole mess…) and he has a position that is very stressful – he has also been using this as an excuse as to why he hasn’t talked to her yet – basically the whole “do you know what I have on my shoulders right now, can I get some sympathy” kind of bullshit. He’s obviously the victim in all of this. And I think it’s so fucking cowardly that he hasn’t told her and even worse – that he becomes angry with me when I keep bringing it up and asking when he’ll speak with her. I’m not completely naive to know that that’s not a good sign, so now I’m at a kind of a loss.
He says he doesn’t want to “destroy” her by telling her, but he expects me to toddle along like everything is fine and like I’M not fucking destroyed every single day. I love this man, but I don’t know what to do anymore. There’s obviously the whole “well, don’t be with him” aspect and I’d be lying if I said I was a fan of how he has handled this situation… but I went from one day thinking that this was the man I was going to marry, to finding out he was engaged to marry someone else and that is just something I haven’t yet figured out how to wrap my head around.
So I guess all of this brings to me to the point of – do I tell this other woman? It has come up in arguments where he says “fine, tell her, I don’t care,” but then I feel as if it would be such a cop out for me to do it – to do his dirty work and save him from having to face her himself. But on the other hand… if I was the other woman, I’d want to know. She deserves to know. And I don’t think either of us should have to continue on in this situation.
I feel so angry, lost and broken and I have no idea what to do or where to go from here. I love him… so damn much… but do I just break it off? Do I give him an ultimatum? Do I tell her myself? Any sort of guidance/advice/suggestions would be very welcomed and appreciated.
tl;dr: Found out my boyfriend of 5 months has a fiance. He has avoided telling her and/or breaking it off with her, even though he swears he loves me and wants to be with me. I’m wondering what I should do and if I should tell the other woman