Red Flag Warning! It sounds like your "boyfriend" needs to stay out of state and if you were to marry him, knowing how he is with your daugther, do you really think things will get better? IMO, they will get worse.
If he is jealous of her now, and only is supportive and generous when she is with her Dad....that should send a message loud and clear, that this man is not great marriage material for you.
It is a heck of alot easier to end a relationship before you get married, rather than wait until afterwards. Hoping he might get better...he won't and then having to go through the stress and expense of another divorce.
Re-read your own post. He is constantly correcting her and you don't like it. You two aren't married and you don't like how he intereacts with your daughter. Do you really see a happy married life to a man you already have concerns and issues with, when you daguther is with you?
Only you can decide. I know what my decision would be.