Akhil Amar emails to note a pertinent discussion of the issue from his book America’s Constitution: A Biography:
In the 1798 case of Hollingsworth v. Virginia, 3 U.S. (3 Dall.) 378, the Court endorsed the permissibility of the practice that had already taken root, under which proposed amendments were not submitted to the president for his signature or veto. Two main theories have been offered to support this result. Some have argued that the two-thirds rule of Article V should be read as creating an implied exception to the usual rule of presentment set forth in Article I, section 7. On this view, since any proposed amendment has already achieved a two-thirds vote of each house, presentment is unnecessary. Others have argued, more directly, that Article V created its own separate higher-lawmaking track above and beyond the presentment clause rules for ordinary Article I lawmaking. On this view, Article V did not envisage any role for a presidential signature or veto in the case of an amendment proposal emerging from a duly called proposing convention; and an amendment proposal made by Congress should stand on the same footing. In 1861, James Buchanan added his name to the Corwin Amendment (which was never ratified), and four years later Abraham Lincoln appended his own signature to the Thirteenth Amendment. On February 7, 1865, the Senate resolved that Lincoln’s signature had been unnecessary and “should not constitute a precedent for the future.”
Apart from the history, one question is whether the original meaning supports this practice. Although I have written several articles on Article V — e.g. see here, here, and here — I have not studied this issue at length. Still, my take is that presentment is required, although I prefer the Constitution without that requirement.
First, Article I, section 7, clause 3 appears to make clear that the proposed constitutional amendments must be presented to the President. It says