I know I am new to this community but I read this entire thread and I feel the need to reply on the children's behalf. I never saw where you mentioned the age of the children but it seems you may have not considered the emotional impact placed upon them for your decisions.
If you decide to see these other men in secrecy, and should it ever come to light, how are your kids gonna react, how will they think of you since obviously your husband has made it very clear that he will be hurt. Are you willing to be the cheater, the wicked witch who destroyed their father? Children may understand after they are adults, but lets be honest here, polyamory is a rare, alternative lifestyle not just anyone can understand, let alone children raised by one poly and one mono. He is their father. How much of his beliefs have been instilled in the kids?
If you get his blessing eventually, and you have visits with the other men, how will the kids respond? What will they think of you then? Remember, your husband fathered these children and I am assuming takes part in the raising of them and therefore may have either picked up on directly or indirectly the mindset of your husband on certain issues, this being one of them. Again, are you willing to be 'less than' in your children's eyes?
I only bring this up because it sounds like your husband is so adamant about this not happening for you and people not finding out that there is no way you wouldn't become the bad guy should the marriage end. If he weren't so against it, then maybe a period of conditioning for the kids would be possible so that the detrimental trauma to the kids wouldn't happen.
My children are grown and they are very aware that their mom is a open-minded and free spirit. Though I have not spoken to them directly about this poly decision my husband and I have made, I do not think they would have a big issue with it because of how I and my husband together have raised them. His children were not raised by us so they might have more difficulty accepting it than mine so before we just surprise them with a second mommy or daddy, we would have to have deep meaningful conversations to avoid any undue animosity or stress to them.
I wish you all the luck in the world. I am saddened that you feel so miserable in your marriage and you shouldn't have to feel that way. However, the marriage is a secondary category for the family unit, of which you are only 1 member. Maybe more thought and a deeper, larger conversation should be had about the responses and feelings of all the other members.