When submitting medical expenses, do I include the total bill or only what I paid out of pocket?

No -- submit the full bill. Way more likely than not, your health insurance contract has a subrogation provivision that allows the health insurance company to recover from the settlement funds paid for your daughter's injuries that part of the bill the health insurance company paid. The health insurance company has to takle affirmnative steps to put you on  otice of its claim.  If they never put you on notice or seek recovery, then they waive their rights.