Hey OP. I've been where you were (except it wasn't even once a month or two - at one point it was an entire year, then got even longer than that.
I didn't "cheat". I remained faithful to him and my vows (I'm not even religious!). And it destroyed me. What's JUST as damaging as not having sex with the person you live is their total refusal to discuss it. It's not a human right but it's a basic need. If he refused to allow you anything to eat other than soup and water, nobody would think that's ok.
When this comes out, and it will, he will not accept any responsibility. You will have cheated on him and he'll be the injured party. He's not accepting responsibility for the pain it puts you in now, so he's not going to at that point!
So depending on how your social life is set up, that could cause massive massive problems in your life.
If you are not sexually compatible, you're not. In that situation it's only ever ok for the person who doesn't want sex, for whatever reason. Someone mentioned asexuality above, but that's not ok either - you shouldn't be forced into celibacy because he doesn't have insight into his sexual drive.
The two of you are best suited to being friends. Thats sad, painful too, but accepting that now BEFORE years more of this or he gets to call you a cheater, means you have a chance to salvage the friendship.
I strongly advocate for you continuing with the FWB but simultaneously not putting your head in the sand, just like he is. Tell him you're going to take a lover, that you're not joking, not threatening, that it's just a fact. If he doesn't accept that then you need to get divorced because while he can continue like this, you can't - and won't.
You can't avoid the pain that's coming one way or another, but how you handle it will determine if you have any self-esteem left (serious discussions with DH or no sex = no self-esteem) and how this impacts your chances of remaining friends longer term as well as your social circle. It's going to be hard, but think of this as him telling you he's gay after marriage and then expecting you to continue living a lie. Nothing wrong with being gay - obviously - but there is in dramatically altering someone else's sexual life again at their will (unless through illness).