The Constitution cannot be amended to change the equal representation of states in the Senate. What would happen if an amendment was passed and ratified that did this? Could the Supreme Court rule that part of the constitution is unconstitutional? Would implied repeal be relevant?
So in the United States, the person who has the authority to recognize that an Amendment has been passed is the National Archivist. He or She would count off that all the requirements are met to proceed to passing and once accepted, the amendment is added.
That said, the most likely course of action is that the Supreme Court would issue a Mandamus forcing the Archivist to reject the Amendment on the grounds that one cannot Amend that into the Constitution... or that the Amendment would only be in effect for the states that voted in favor of the Amendment (since they consented) and any new states that came aboard following the the ratification of this proposed amendment.
Article V does not say that Article V cannot be amended, so an amendment could be passed that deletes the text "and that no state, without its consent, shall be deprived of its equal suffrage in the Senate". This does not remove California's representation in the Senate (etc) therefore California does not have veto power. SCOTUS has not previously ruled that this clause cannot be amended out of existence, or that there is any part of the Constitution which cannot be amended or which must satisfy criteria other than those stated in Article V. Of course, there does exist an ideology to the effect that Art. V esp. the last clause is sacrosanct, but that can be firmly asserted only when SCOTUS so rules (a ruling that might be later overturned, depending on the influence of "original intent" on the court at the time). If you compare the language of the "1808" clause with the "equal suffrage" clause –
no amendment which may be made prior to the year one thousand eight hundred and eight shall in any manner affect the first and fourth clauses in the ninth section of the first article
no state, without its consent, shall be deprived of its equal suffrage in the Senate
the former clause clearly has a broader scope (affecting in any manner). The Framers had the option of stating the suffrage clause similarly ("shall in any manner affect the equal suffrage of a state..."), but did not. What you don't say in writing laws and constitutions is good textual evidence for original intent.
Anyhow, it would be up to SCOTUS to rule on this question.