How can I claim head of household despite unmarried spouse claiming our child upon non mutual agreement and my higher income providing the majority support of the home?

Please provide some more details on "unmarried spouse" and your overall situation. Are you legally married, or just living together in the same home with a child in common?

If you are married, the only legitimate filing statuses are Married Filing Jointly and Married Filing Separately. 

If you live together  and are unmarried you do not have a "spouse". Only one parent can claim the child and ALL the benefits.

If you CANNOT agree who will claim the child (why are you living together then?) the IRS will decide if yiu both claim the child.

(If you are separated and do not live together then there are different rules).

Qualifying Child of More Than One Person

[Quote from IRS Publication 501]
Sometimes, a child meets the relationship, age, residency, support, and joint return tests to be a qualifying child of more than one person. Although the child is a qualifying child of each of these persons, only one person can actually treat the child as a qualifying child to take all of the following tax benefits (provided the person is eligible for each benefit).

1. The child tax credit or credit for other dependents.
2. Head of household filing status.
3. The credit for child and dependent care expenses.
4. The exclusion from income for dependent care benefits.
5. The earned income credit.

The other person can’t take any of these benefits based on this qualifying child. In other words, you and the other person can’t agree to divide these tax benefits between you.

Tiebreaker rules.

To determine which person can treat the child as a qualifying child to claim these five tax benefits, the following tiebreaker rules apply.

1 - If only one of the persons is the child's parent, the child is treated as the qualifying child of the parent.

2 - If the parents file a joint return together and can claim the child as a qualifying child, the child is treated as the qualifying child of the parents.

3 - If the parents don't file a joint return together but both parents claim the child as a qualifying child, the IRS will treat the child as the qualifying child of the parent with whom the child lived for the longer period of time during the year. If the child lived with each parent for the same amount of time, the IRS will treat the child as the qualifying child of the parent who had the higher adjusted gross income (AGI) for the year.

4- If no parent can claim the child as a qualifying child, the child is treated as the qualifying child of the person who had the highest AGI for the year.

5 - If a parent can claim the child as a qualifying child but no parent does so claim the child, the child is treated as the qualifying child of the person who had the highest AGI for the year, but only if that person's AGI is higher than the highest AGI of any of the child's parents who can claim the child.

Subject to these tiebreaker rules, you and the other person may be able to choose which of you claims the child as a qualifying child.
[end quote]

https://www.irs.gov/publications/p501#en_US_2018_publink1000220917