The instructions state that you are to enter the settlement date in the appropriate section of the HUD 1 or 1A. If the dates differ I cannot opine as to how your examiner will view that.
It would, IMO, be a technical violation.
What I would be concerned about if the settlement statement and all other loan documents are dated the 10th and the borrower signed and dated everything on the 12th; if the transaction is subject to rescission your rescission begins on the 12th not the 10th, and your rescission notice would have the wrong dates therefore being inaccurate.
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The opinions expressed are mine and they are not to be taken as legal advice.