If the delegates are elected to this on a platform of not having to wait twenty years for the changes to pass 38 state legislatures, could they constitutionally waive that necessity?
Nobody knows. The last convention ignored the ratification requirements of its governing document, this one might too. The Court probably wouldn't touch anything about it as it'd be political matters and the Constitution doesn't set limits on the convention's authority. It'd be a giant constitutional crisis because once a convention is called it'd have as much political legitimacy (or more) as Congress or the state legislatures would.
Either way, I'd remind you that another option exists in the Constitution besides getting 38 legislatures to agree once the amendments are written out: Congress could elect to call state conventions to ratify or reject the amendments instead of going to the legislatures. That'd almost certainly be a faster process.
Sure, but the Convention's mandate would already be quite large. I can't imagine the military overriding the Convention. Furthermore, can Congress really claim authority over the Convention, or would we see the two engage in a proxy legal war?
This is the fundamental problem. It is highly likely that once a Convention is in session, they are effectively re-founding the country and no legal precedents that predate the Convention apply.
