She has chronic alcohol and drug problems, he said.
The father has a conviction for a sexual offence against a minor and has no guardianship or access rights.
Home for Life status means the Baylisses, as guardians with the birth mother, must consult her on all "important matters affecting the child".
The couple are 39-year-old teachers but Mr Bayliss is out of work.
"I'm looking out of town for work at the moment, but if we had to relocate, there is the potential for it to go to court," he said.
Mrs Bayliss is an American citizen so Mr Bayliss has been looking for work in the US, but they would not get a residence permit for the child unless they legally adopted her.
He said they took the child to meet her birth mother several times a year. "It usually happens because occasionally [the child] will ask to 'see my tummy mummy'."
He said the mother told them repeatedly she would consent to adoption, but after they arranged for her to see a lawyer she would not consent. Another lawyer said it could cost $15,000 to $20,000 to seek a court-ordered adoption.
"The lawyers are saying you'll have an easier time getting a law change to allow you to adopt a child from foster care than using the Adoption Act," Mr Bayliss said.
Social Development Minister Anne Tolley said the law had changed already. Last year's Vulnerable Children Bill created a new "special guardianship" order by which a Family Court judge will be able to share rights between guardians from July next year.
The Bayliss' petition is online at: facebook.com/Righttobelong
Between two stools
Foster care: Child, Youth and Family is the child's legal guardian and pays foster parents allowances of $146 to $204 a week for each child, plus clothing allowances of $78 to $132 a month, depending on the child's age.
Home for Life: Foster parents are legal guardians, usually with the birth parents, and get an unsupported child benefit at the same rate as foster parents but with no clothing allowance. They also get a $2500 lump sum at the start.
Adoption: Adoptive parents are sole legal guardians. The only support is family tax credits of $64 to $102 per child.