A Google search shows others reporting this calculation error made by the IRS. Apparently the IRS was including the net profit as a negative amount on Form 8960 line 4b without first including it on line 4a. Had it been excluded from both lines as was done with tax returns for years prior to 2022, there likely would have been no net error in the calculation of the NIIT.
"Given that the error was on the IRS's side, is it possible to request a refund for the TurboTax software I used for that year?"
I see 2022 TurboTax correctly preparing lines 4a and 4b of Form 8960, so seems that there would be no justification for an accuracy-guarantee claim against TurboTax. The accuracy guarantee only covers penalty and interest anyway, so if you pay within the time allotted by the IRS there would be nothing to recover.
It appears that the IRS changed the reporting requirements between 2021 and 2022 to include Schedule C profit on line 4b and cancel it out with an entry on line 4b (maybe to be able to treat passively earned Schedule C profits, which is probably unusual, as investment income), so it's not clear when this got implemented in TurboTax or when the error was corrected in the IRS system. There is this post about the issue in these forums:
https://ttlc.intuit.com/community/taxes/discussion/form-8960-error-saga/00/2875681
If the IRS originally "corrected" your 2022 refund based on their erroneous calculation, inappropriately increasing your refund, then that is justification for the IRS to ask you to now pay back to the Treasury the amount inappropriately refunded.
