Double-check the handbook, expressly checking to see if "Paid time off (PTO)" is expressly excluded from notice periods.
If use of PTO days in notice periods is not expressly and explicitly disallowed, it is unlikely that excluding PTO from notice periods is actually part of the terms and conditions of your contract with your current employer. Their claim is, therefore, unlikely to be well-founded, and so is unlikely to be enforceable.
Additionally, and to re-emphasis how unlikely it is, notice periods of the form "nn months / nn weeks" are by default, referring to elapsed calendar periods and not the number of working days normally equivalent to that elapsed calendar period, assuming that all normal business working days in the elapsed calendar period are utilised as working days by the employee handing in their notice of departure.
And given how convoluted one sentence is compared to the other, you can see why the default is "elapsed calendar period" - a three-word phrase, and not the 38-word alternative.
That having been said, a) I am neither a lawyer nor a Human Resources specialist, never mind one with current professional qualifications for operating in your current jurisdiction. Take these words above with a pinch of salt. b) There are times when being a stickler for the written rules can be inappropriate or counter-productive.
How important is it for you to leave on glowing terms with the current organisation? And by comparison, how important is it to you to leave on precisely that date?
Negotiation and discussion may be a better tactic than standing resolutely upon the letter of the law.
"I'm afraid the employee handbook does not exclude previously-approved PTO from notice periods. It might be possible to for me to stay on for one additional working day if [some condition, or some additional benefit to you]. Is that what you would prefer?"
