How can I get a Green Card and the custody of my child?

It sounds like you need to schedule a free consultation with an experienced immigration attorney. There is some potential (based on extremely limited information) for a U visa petition, but there may be other options as well. The problem is that entering on a fiance visa and then failing to marry will likely result in removal proceedings. Speak to an attorney as soon as you can.

Unless you get married to your fiance, there is no other way for you to get your green card even if he is abusive toward you. As for the custody of your child, you will need to speak to a family attorney to determine your rights and obligations as a parent. contents of this information is strictly prohibited. If you have received this e-mail message in error, please immediately notify the person identified as the sending person by reply e-mail or call the telephone number set forth above.

If he will not marry you, you cannot get a green card. Also, if you overstay the K-1 visa I-94 and do not marry, you will NEVER be able to legalize your status within the US through ANY OTHER MEANS. So, I would strongly suggest that if you are not going to marry him, depart the US in a timely manner so that it does not impact on your future ability to immigrate. Please let me know if you have any additional questions or how I can be of further assistance to you in this matter.