"So it seems that the $1600 does not have to be prorated because we have HDHP family coverage and share the $7100 HSA contribution limit (for 2020)."
The $1,600 is his part of the $7,100. For all we know, it could be allocated to him in the first few months of the year when he was still eligible.
Since this limit calculation is done on an annual basis, you are free to allocate as you see fit.
However, the $1,000 "bonus" belongs to individual's HSA, and the IRS says that the "bonus" has to be pro-rated just like the main amount. Since only HIS eligibility applies to his HSA, the $1,000 has to be pro-rated. As Dawn reminds us, the Family coverage is good for the whole year, because you had the Family coverage for the whole year.
I wish that I could point out an easy place to find this. However, IRS Publication 969 does not cover many unusual situations. You could also look in the instructions for form 8889, where there is some discussion that suggests that this is how Family coverage should be applied.
But you would also have to read the various IRS Notices that have been issued over the years, for discrete Q&As that often provide much more insight. IRS Notices are typically read only by tax professionals, but they do tend to fill in the gaps in the official publications and instructions.
Bear in mind that as a TurboTax user, you benefit from the Tax Accuracy Guarantee, so if the IRS ever questions how your return is done, then you can apply under the guarantee, and the group in charge of that will show why our calculation is correct. You are not going to be left out on a limb.
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