Yes, the HSA is supposed to pay for medical bills that don't get reimbursed.
There are a couple of ways to approach this.
1. Ask your HSA plan administrator to treat the amount reimbursed as a "mistaken distribution". In this case, you will send them the money that insurance covered and they will issue you (eventually) a corrected 1099-SA. Note that the plan administrator is not required to honor your request.
2. An easier way to approach this is for you to use the insurance money to pay medical bills for a while instead of using your HSA until the insurance reimbursement is exhausted. Since the insurance money is also tax-free, this is the same effect as using the HSA, without the paperwork.
