Joint custody

Paying the health insurance for the child is not relevant if you are not the custodial parent.

A child can only be claimed on one return each year. The IRS tie-breaker rules for who can claim the dependent child are as follows:

• The parent, if only one of the persons is the child's parent,
• The parent with whom the child lived the longest during the tax year, if two of the persons are the child's parent and they do not file a joint return together,
• The parent with the highest AGI (Adjusted Gross Income) if the child lived with each parent for the same amount of time during the tax year, and they do not file a joint return together,
• A non-parent, if no parent claims the child as a qualifying child although he or she may do so and only if the non-parent's AGI is higher than the highest AGI of any parent who may claim the child, or
• The person with the highest AGI, if none of the persons is the child's parent.

From IRS Tie Breaker Rules

Assuming the child spent 50/50 time with each of you, the parent with the higher income (AGI) would be the parent eligible to claim them as a dependent.