First, I would withdraw the application.
Second, I would get a paternity test.
If she is already doing these things, it will be much worse once she is a citizen and doesn’t need you. She can keep the child from you here as well as there. Don’t let her blackmail you.
Petition for full-custody, if possible, after things have settled down. If the allegations are true, she’s going to be an unfit mother regardless.
Best of luck.