Friends with Best friends ex

So I'm stuck in a strange position I'm friends with my best friends ex we've gotten pretty close over the summer since he was the one I talked to when my BFF was studying abroad for most of the summer and we worked together. My BFF wants me to stop talking to him "cut him out" because of their last fall out but I don't want to am I a bad friend if I still hang out

With him ?

Full story -> My best friend dated this guy I've known since elementary school. Let's call him Mike. We drifted apart since then but we got close again when my best friend was dating him. We would go on double dates with my boyfriend at the time and play video games together. They had a bad argument, both of them have strong personalities, and split up after two months . After some time went by Mike and my BFF started talking again and decided to be friends. At the beginning of the summer we started playing video games together again with more people everything was fine until my BFF went on vacation. Before she left it seemed like she and her and Mike were on good terms I even thought they might get back together but when she left they stoped texting as often, they had another argument and were in different time zones. The group that was playing video games fell apart and only me and Mike were left I didn't think it was weird at the time because it was just video games. After a month My best friend came back and things were kinda weird between her and Mike and she would always make comments about me being buddy buddy with him. when things were weird between them she would usually ask me to text if and ask him questions I was the middle man and I realize now I shouldn't have gotten involved but now it's too late I'm close friends with Mike now and I don't want to end the friendship just because she wants me to however, I feel horrible when I hang out with him since I know she wouldn't like it.