The IRS defines "Family HDHP coverage is HDHP coverage for an eligible individual and at least one other individual (whether or not that individual is an eligible individual)." (see page 4 of Publication 969).
What you are telling me is that you are paying the higher premium for a Family plan by keeping your daughter on your policy, even while your daughter is paying her own HDHP premium.
I don't see that there is any conflict here in terms of HSA contributions - since both coverages are HDHP plans, there is no disqualifying coverage for your daughter.
Note that, of course, you know have, in essence, primary insurance coverage and secondary insurance coverage, not simultaneous coverage. That is, you now have two deductibles that you two have to satisfy, unless you make a point of making all claims against only one policy. This is why most taxpayers would have changed to Single coverage.
If I am not mistaken, your daughter must have been less than 26 years of age in 2019. Otherwise, if she was not your dependent, I don't see how she could be added to your insurance. Correct? This was a function of the Affordable Care Act, that you could keep children on your health insurance policy up to age 26, even if they were no longer able to be claimed as a dependent.
Eventually, this means that you will have to drop her off your policy, right?
In the meantime, the example that the IRS gives under step 4 on page 4 of the instructions for form 8889 (note, this is for 2018, but so far as I can see, these rules have not changed) implies that your daughter can have a separate HSA contribution limit once she has her own HSA.
OK, your daughter who is not your dependent got her own HDHP coverage starting July 1, 2019. Now the question is when she got her HSA.
She can make contributions in early 2020 (before the filing deadline) towards her HSA for last year, because she was covered under an HDHP policy all year. In terms of her own HSA contribution limit, yes, ordinarily, she would be limited to 1/2 of the annual HSA contribution limit for a Single policy ($1,700, which is 1/2 of $3,400), but she is able to invoke the "last-month rule".
The last-month rule allows a taxpayer to use the full annual HSA contribution limit, so long as the taxpayer has HDHP coverage on the last month of the year (that is, on December 1st). Thus, your daughter's limit for 2019 is $3,400. The only catch is - as noted above - that she will have to stay under HDHP coverage for all of 2020.
As you realize, she can contribute to her HSA in 2019 up until the due date of the return - even though her HSA did not exist yet, because she did have HDHP coverage).
However, she cannot reimburse herself for any medical expenses that she incurred prior to the creation of her HSA. Thus if she did not create the HSA until November 1, 2019 (made-up date), she could not reimburse herself from her HSA for any expenses that she had incurred prior to that date.
OK, I apologize for the lengthy response, but this is an unusual case that the IRS does not seem to have specifically addressed, and it is caused by the HSA code (Section 223 of the Tax Code) being adopted well before the Affordable Care Act, which changed the rules to allow children who are not dependents to be still carried on your health insurance policy until age 26.
I trust that this all makes sense.
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