No, it would not double count the data. If you are using TurboTax Online, you'd input the entire K-1 for the first K-1 entry, along with Box 16/K-3. The remaining K-1 entries will only have entries for Box 16 for the K-3 data.
If you are using TurboTax Desktop, you can follow the information here to generate new K-1 Worksheets in Forms mode.
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