"Maybe I should move out"

"Yes, maybe you should."

He's just trying to guilt you into staying quiet. Don't stand for it. Next time, call his bluff.

Any guy who threatens to break up with you or have you moved out over any small dispute is not the kind of guy you need in your life.

This. So if you try to address any issues, this is going to be his go to response.

Next time he says that say 'Well, if you are going to say that everytime I bring up some concerns and won't discuss them like an adult, then yes, maybe you should move out.'

You're needs and feelings will slowly be dismissed over time because eventually you won't even want to bring them up. It's just another deflection tactic like when you come to someone with a concern, the next thing it is turned around and you are discussing the problems that person has with you or what you've said or done.

Don't stand for it. It might not seem like it now, but it might be better that he leaves if you can't talk about things.

I agree with the other posters.  They are all right.  Call his bluff and get him to move out.  He's using this threat to manage you.

Think about it this way:  you wouldn't stay at a job if they threatened you with termination everytime there was a little stress.  Why would you accept less in your personal relationship?

I see it from a different angle.  He wants to break up, but doesn't want to be the bad guy.  He makes the threat hoping you will call his bluff, and kick him out.

Call his bluff. 

That is emotional blackmail and Gaslighting. Next time he is being a d**k, call him on it. If he says "maybe its best if I move out" - say 'Yep, I agree. My job is to have a home where my daughter feels safe and loved. You don't get to take that from her. You don't get to have special rules for your little snowflake and treat my kid like crap."

He may be an improvement over your exH, but he is definitely NOT "a lovely guy". As mentioned above, he's a bad parent and an emotional blackmailer.

And he most certainly IS a lover of drama. "...maybe best I move out and we break up etc..."

There is no maybe. He SHOULD move out. I feel sorry for your daughter. His behavior is causing her unneccessary anxiety. You are showing her that a bad man is better than no man. 

You can hire someone to help you around the house. This guy isn't worh it.

Quit tolerating his bullshit threats to move out and break up.

"If you don't figure out your issues with my daughter who makes your daughter look like a picky drama queen don't waste your breath with the threats to leave and don't let the door hit you or your kid in the ass on your way out of our lives."

Why tolerate this immature idiot and his bullshit middle school games?

In a relationship you have to be able to talk about problems or even argue without the other person threatening to leave. 
Too many people use the phrase "I don't like drama" as an excuse to avoid talking about problems (or their bad behavior)

Its true you should try to talk about problems in a constructive manner but if you can't have an argument with your SO, that's a really weak relationship. There's bigger problems here than nitpicking.

Let him move out.

Neither you, nor your daughter deserve this treatment.

How long have you known your bf? How long were you together before he moved in? I think it is always best to wait a very, very long time before moving anyone in with your kids. Anyone who wants to do so quickly probably has some issues (unstable, nothing to lose, psychological problems, etc). Most people who have their life together don't want to jump into moving in with someone who has children and will wait it out until most of the issues are sorted which takes time.