My feeling is that not all relationships are inherently alike. Each relationship dynamic is as individual as the people involved. The important part is to find what works best for the relationship. For instance, I have friends (gay & straight) who have said they know their S.O. messes around and as along as they keep it DownLow they are fine; and they act accordingly, as well.
On the other hand, many are traditional and do not want to "share" their S.O. with anybody else, nor feel the need to have outside interests.
Personally, I want/have an open relationship. Communication is the key and consensual activity is agreed to by all parties.
Plenty of married men (with wedding rings) and men (gay/straight) in relationships, have anonymous sex hookups. Some say their S.O. know, others are "cheating".
I try not to judge as I recognize I am no position to do so, therefore: To Each Their Own.
The best thing is to do your best to be honest to your own convictions. If your friend's marriage changes things for you and you are uncomfortable, placing yourself in a position of regretting future sexual connections does not seem in your best interests. If you can live with yourself if his marriage ends due to his wife's discovery of your sexual connection, then proceeding until apprehended could work. It does seem, however, that she could discover his bisexuality even without you two continuing to hookup and she may still end it. That may be something you want to come to terms with also.
This is your decision and a number of posters have given some really good advice. You have to decide what is best for you, what you are willing to live with and what you will not live with.
I understand how your friend's vows in marriage have affected your sensibilities to your sexual acitivies. You sound like a caring person and someone who is torn between your own sense of fairness and your desire. That is a very difficult place. I wish you, your friend and his wife the best outcome. Good Luck.