The Congress may determine the Time of chusing the Electors, and the Day on which they shall give their Votes; which Day shall be the same throughout the United States.
No... that refers to a uniform protocol, a run-up time limit. It does not mean you have a single set day to ‘choose’ electors. It means there would be a cut-off day in which you must have completed the appointment/validation and resolved any challenges to any appointment by legal remedy of all electors that qualify to be an official delegate that has been given authorization to cast a vote in their respective state capitals that will be the same day for all states.
If a state legislature has not yet chosen by a statutory date(a prescribed deadline) and validated all entitled delegates ... then any delegate that failed will not be included in the vote tally once the votes have been cast in the state capital.
This does not mean a one-time shot. It simply means a set of rules in the processing of delegate votes.
State legislature delegates CAN challenge the joint session vote in congress by petition to amend their delegate votes. It is not cast in stone after the meeting of the delegates on election day in the state capitals.
So no... there is no constitutional same-day choosing and you have not shown me that it is imposed by the constitution... it’s not.