Tiebreaker rules only apply if more than one person actually claims the child.  If they agree for who claims the child, the Tiebreaker rules don't apply.

So if they agree, they can choose who claims the child.  It sounds like the father is the best option in this scenario.

However, because the mother makes more income, it sounds like she may be able to claim Head of Household if she claims the child.  But if the father claims the child, the mother would not be able to claim Head of Household.