I'd suggest trying to figure out why you're dreading it without having ever done it.
He is a father. He already HAS "that connection" with another woman. You just preferred to ignore it.
Just because he files doesn't mean he'll win anything. You may have nothing to worry about.
And, even if he gets to see her, it is in YOUR family's best interest to be as open and accepting as you can be...OR you run the very real risk of having to share your child with him, too.

You only have one option if you are staying with your husband. Accept his kid. You do not have to try to be her mother, in fact, that is the worst thing you can start off doing. Do not try to please him or impress him by doing for her or trying to mother or parent her. She is his responsibility and you will only grow to resent her and him if you put it on yourself to raise her.

All you have to do is treat her with dignity and respect, be an authoritative figure in the home, but not a parent. Like if your kid had a sleep over or you had a niece. If you are making every one lunch, she is included, however, Dad is responsible for all parent-related issues. If he wants you to watch her, he should ASK YOU and not just leave her with you expecting you to fill the mothers role.

You need to set boundaries from even before she gets there. Tell him you are willing to be supportive, but you have to establish boundaries and expectations of you, him, and her to ensure you all can remain a family together. Tell him you expect him to manage her behavior and to set rules for her that he follows through on and in turn, you will treat her with kindness and be warm to her, but do NOT feel like you have to LOVE her or pretend to love her, how can you love a stranger kid? That is what I think screws up most stepmothers, they feel like they have to be motherly and love the kid and they grow more and more resentful when they get frustrated over the kids disrespect or behaviors.

Any way you and your husband could see a counselor that handles stepfamily situations to help you guys get on the same page and understand each other, what you both want and expect and what you both need, a little before she gets here?

you don't say what your BF reasons are for filing for custody? It sounds like he must not see the kid very often if she lives in another Country, so why in the world would he just randomly file for custody now?

I guess I wouldn't get too worked up about it, because unless mom is way beyond bad, I can't see any court awarding him custody.