It's your apartment, your life, no one can live it for you. Make your own decision. If you want her to live with you, ask her. Your boyfriend may object, but ultimately you belong to no one.
It seems obvious to me, from what you've written, that he feels a certain sense of ownership over you: surrounding your living situation, giving you "permission" to be with women, etc. He didn't feel comfortable with you having another partner and brooded about it, tried to be involved with other women, but that didn't work. Interesting that the sex started fizzling out after that, but possibly not a coincidence. I'm not surprised, actually. The fact that you have less interest in your boyfriend could be that you are sensing in him his ambivalence (or discouragement? resentment?) toward you possibly over having this new, loving relationship in which he feels excluded. Sounds like plain old jealousy to me.
It may very well be that your relationship with him is coming to a close. Some relationships are finite and not meant to last forever. They have a beginning, middle, and end, and that doesn't negate the good times. But it also could be that he is envious and can't deal with it and you're picking up on it. Some people deal with difficult feelings by basically having a sort of tantrum, which isn't very attractive - especially when being with your girlfriend feels so joyous and free. I think you need to communicate and see if there are any difficult emotions underneath it all, on his part, which are sabotaging what you have.
I would also reevaluate how much say he has in your life. Is he a partner with whom to move forward in life, encouraging you to grow and be yourself, or a dictator wannabe who pouts when you find satisfaction and joy with other people?