My fianc? (33M) refuses to invite his ex/my friend (30sF) to our wedding; he went behind my (31F) back and lied to her and yelled in my face about it

Before I (31F) met my fianc?, I'll call him Jake (33M), he was engaged to a woman I will call Sharon (30sF). They were together for 6 years total and broke things off very amicably. They have remained friends and actually made a pact that they would attend each other's weddings later in life. Sharon got married a few years ago and we were in attendance. I love Sharon and we've actually formed a friendship of our own since I was first introduced to her.
When Jake and I were sorting out the guest list, I automatically put Sharon's name down. He crossed it out without telling me. When I asked him about it, he said he just didn't want her there. I asked if anything had happened and he said no, he just did not want her in attendance. I asked Sharon if they had recently fought or anything and she said no, they actually hadn't really talked in a few months. I did not disclose why I was concerned that they had. The next day, Jake told me he did not want Sharon at our wedding and that I shouldn't pester her about their relationship. I left it alone, out of respect to him.
A day after that, Sharon sends me the following text (copy pasted):
I never have been upset with Sharon, so I called her and asked what was going on. According to her, Jake sent her a text saying that I was mad at her and did not want her in attendance at the wedding. In his text, he was "really pushing" for Sharon and her husband to attend but that I had shut him down. I was so upset that he threw me under the bus and potentially damaged my relationship with her. I let her know that was not the case and that she would be getting an invite.
I confronted Jake about this and told him that if he doesn't want me to meddle in his relationship with Sharon, he shouldn't meddle in mine. He got angry and told me that he couldn't trust I wouldn't invite her behind his back.. I told him off for being so untrusting of me when I have given him ZERO reason to feel that way and reminded him that MY relationship to Sharon is MY business. I said I really wanted to invite her and would like to understand why he didn't and he got in my face, less than a foot from me, and shouted in my face "NO" like a toddler throwing a tantrum. I told him I would not be treated like that and made him leave the house. When he came back, he apologized to me for yelling and told me to let it alone. Since then, we have been walking on eggshells around each other and have not even said the word "wedding" to one another.
This is no way for us both to live but I am at my wit's end. I was ready to leave everything alone and then he went and spread lies to Sharon, nearly damaging my relationship to her. He screamed in my face and disrespected me as a person. I'm truly wounded by this all. He has never been the most sensitive and has always kind of been a bit of a caveman, but he is at the very least courteous and kind. This is NOT the man I said yes to. This is NOT the man I want to walk down the aisle with. But I don't want to throw away FOUR YEARS of a relationship over a petty argument.
I have no family and I don't trust that my friends would be unbiased, so I need some help Dating.mobi. Share your wisdom. What am I meant to do with this man? Where do I even start with this? Do I go back and talk to Sharon about this? How can I get him to talk about this more with me?
TL;DR: My fianc? did not want to invite his ex to our wedding so we didn't. He then lied to her about how I felt towards her and screamed in my face when I confronted him on lying to her. I have no idea what to do here. Where do I start?


Source.