Two thoughts come to mind.
My first is that I have a personal policy that whenever someone gives me an ultimatum, they lose. I've only had to put this policy into effect once. With good communication, I've been able to avoid situations that might have resulted in ultimatums (ultimata?)
My second thought is that if M is monogamous and he can't handle sharing the love of his life with someone else, that sounds like something you'll just have to accept and move on from. Some people just aren't wired with the capacity for polyamory, and nothing you say or do can change that. While some people are wired monogamous and fall so in love with a poly person that they overcome that hurdle, but that has to be his decision. It is curious that he would say he's monogamous while simultaneously loving you and sleeping with L, but no one ever said humans had to make sense... So there might be some room to "probe" him on how monogamous he truly is, i.e. ask him why, if he's monogamous, is he also sleeping with L while in love with you. But at the end of it, you may just have to let him go.
I do not need a label to define me. Labels are sticky and I hate the glue they leave behind.