I don't think you love your fiancé as much as you say you do. You get irritated when she touches you in public? So you can't hold hands when you are out? Or lock arms? Or hug? That makes no sense, of course you want to look like friends but you are also supposed to be lovers. Your lover should BE your best friend. If I held my man's hand, and he jerked it away and scolded me because he felt embarrassed, I would tell him to **** off and dump him. This friend of yours? I just got done reading and replying to a situation similar and you know what happened there? Her boyfriend screwed his supposed female friend. You are a man, and this debt you feel you owe is garbage. The fact is, when you decide to marry someone, you have to give up some things, things that will ruin a relationship. This friend is something you need to let go of respectfully but I don't even see that happening. You will lose your fiancé if you keep of this tug and pull game between you and your friend. I have had to get rid of male friends before but they gave me space with my previous boyfriends. They didn't want ME to have problems with my guy and kept in contact with me to make sure I was doing ok, that is about as far as it went, and when I broke up with my ex's, they were in my lives again supporting me and my decisions. You can't convince someone that someone else is just a friend ESPECIALLY when they act like that. I have stayed away from some of my guy friends because I didn't want them to have problems with their girlfriends. It's called respect, and apparently this friend you owe so much to, doesn't have any respect for you what so ever. So keep it up, and eventually you will be alone.
“I was never really insane except upon occasions when my heart was touched.” ― Edgar Allan Poe
Wish for a pile of shit to turn into gold hard enough and guess what? It's still a heaping pile of shit.