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

Every Order, Resolution, or Vote to which the Concurrence of the Senate and House of Representatives may be necessary (except on a question of Adjournment) shall be presented to the President of the United States; and before the Same shall take Effect, shall be approved by him, or being disapproved by him, shall be repassed by two thirds of the Senate and House of Representatives, according to the Rules and Limitations prescribed in the Case of a Bill.

Certainly a proposed constitutional amendment is a vote “to which the Concurrence of the Senate and House of Representatives may be necessary.”  For a different view, see here.

Second, I don’t see anything in Article V that would lead to an exception.  Structurally, the requirement that a proposed amendment receive a two thirds vote does cause one some pause.  But requiring the Congress to re-pass a vetoed amendment by two thirds is not purposeless — sometimes members of Congress change their votes in response to a presidential veto.   If there were some ambiguity in these provisions, then this structural argument might persuade me, but I don’t really see an ambiguity.

One possible way of finding an ambiguity would be if the terms bill order, resolution, or vote had limited meanings that did not cover proposed constitutional amendments.  But I doubt it.