I think your spouse is making a mistake. It's one thing to enter the U.S. as part of a bona fide marriage, get divorced, and try to get permanent residency. It's quite another to fabricate a record of cohabitation following separation. While your spouse may have married you for love and entered the U.S. for all the right reasons, if the USCIS catches wind of the fact that you and she are lying about the status of your marriage and living arrangement she is apt to find herself both without a Green Card and on her way back to her nation of origin. She should consult an immigration lawyer.
What you should do is discuss your situation and goals with a divorce lawyer, as that's where things are headed. You need to decide if you want to try to get custody and, if so, whether you can demonstrate to a court that you have a suitable home for the child and the ability to support your household. The longer you choose to stay outside of the home and to make her the primary caregiver, the less likely it will be that you'll get primary custody.

