Edit:   They are unmarried. They live together. He is the higher wage earner. She can claim the child by agreement or divorce decree. Her filing status is single.

fwiw, she could be eligible for the eitc.  It is much simpler to default to the higher income earner than go through the gyrations of trying to prove something that could take hours to figure out and justify. The IRS should accept the higher income earner. as someone in this thread stated, with a dependent the difference between single and hoh is probably minimal.