Upvote:5
Article 2 of the United States Constitution states that:
The executive power shall be vested in a President of the United States of America. He shall hold his office during the term of four years.
So yes, it would indeed have required a constitutional amendment prior to the 1864 election for the President's term-of-office to be extended beyond four years.
This would not have been entirely without precedent. The government had previously shown that it was willing to amend the Constitution to address unanticipated problems. For example, the Twelfth Amendment was passed in 1804 to address problems that had arisen with the original procedure contained in Article II, Section 1, Clause 3 of the Constitution during the Presidential elections of 1796 and 1800.
As you observed in the question, Lincoln rejected the suggestion from his advisers that Article 2 of the Constitution should be amended. It has not been amended since Lincoln's time, so any future proposal to extend the President's term-of-office would also require a constitutional amendment.