You can establish your paternity under Florida law either by affidavit (available from your county department of health or the Department of Children and Families) or through a legal action (
form and instructions, PDF format).
To get the custody arrangement to "stand firm" your best bet is to have a court order defining the parenting time arrangement. You may wish to consult a local family lawyer. If you bring a court action you can get a custody order once you have been determined to be the father.