I know you say that there is no sex involved in your relationship with you friend. But I'm going to approach your problem in that manner anyway. You will understand why later.
Everything in the male ego has to do with sex and the number "3". One Male + two Women = Sex. Regardless of weather or not the two women involved are satisfied. When a man hear's the word "lesbian" the idea of sex automatically comes to mind. (As if the two woman really want anything to do with him.) But the male ego tries to but himself in to situation anyway. Two lesbian females + One Male= Sex. In the event that there is only one Male and one female the Male will be thinking about some other Female besides the one he's with. One Male + One Female (+ one Female)=Sex. Men don't have relationships, they have sex. While for the woman it is all about the relationship. Men only tolerate the idea of the relationship because it is required by the female in order for them to have sex. Men think about sex all the time, consciously and unconsciously. The average male gets an erection 11 times during the average day.
You are in a unique position because you can utilize your female side to understand what you fiancee is feeling. It's not so much about you having sex with your friend that bother's you fiancee. It is the whole relationship issue!
Your fiancee see's your friend as a threat to a committed relationship with you.
Your friend see's your fiancee as threat to her relationship to you.
And you, because of your male ego, actually think you should be able to have it both ways.
That leaves you only two choices.
(1) Dump the friend and keep the fiancee.
(2) Dump the fiancee and keep the friend.
Of course the male ego likes to think there is a third option.
(3) Have a threesome. (Just ignore that one.)