Let me answer you in this way: yes, a state could authorize it's delegate to do nearly anything it wanted to, but a vote to ratify even if offered at an Article V Convention would be utterly without effect since an Article V Convention has NO POWER to ratify Amendments. None. It can only PROPOSE them; to be ratified they must be acted upon by each of the States, individually.
Let's try to scale back the nonsense, OK.
It wouldn't be the "convention" voting to ratify it would be the state delegates that had authority to do so voting to ratify. The constitution says nothing about where or when the ratification vote has to take place. If enough state delegates (3/4 of all states) with the legislative authority to do so voted to ratify the amendments just minutes after they were approved by 2/3 of the State delegates then it would be a legitimate state ratification. Nothing in the constitution would prohibit this.