First: when you went to court in regard to the custody hearing, did you object to this judge hearing it? (Did this judge actually do anything in the criminal case other than accept the guilty plea?) If you didn't object then, you don't get to decide AFTER you've seen his ruling that you want to object NOW.
You're going to have the same problem with this judge, or any other judge, because the fact that you had a criminal conviction for a sex offense involving a minor, which was apparently AFTER you were the father of this child, is going to be admissible in a custody hearing, and will probably have a significant effect on the custody order any judge will make.
So, from the time of your daughter's birth, to one year ago, what did your parenting skills look like? How much did you participate in your daughter's life? As much as anything else, this will be important information to the judge. What did the judge do about the visitation schedule?