Answer:No. The Founders made this very clear. Once 34 states apply, Congress has no discretion whether to call a convention and no control over the delegates (see Federalist No. 85, see paragraph beginning “In opposition to the probability…”). George Mason proposed to add the Convention of States provision to Article V because he thought Congress had too much control over the amendment process. The Framers unanimously agreed with him. It makes no sense to interpret Article V to give more power to Congress, when the whole point was to take power away.
This claim that Congress gets to choose the delegates also goes against common sense. Just because one party “calls” a convention, doesn’t mean it gets to choose the delegates for the other parties. Think about it. Virginia called the Philadelphia Convention of 1787. Did it get to choose the delegates for Massachusetts? Of course not. Massachusetts did. Each state chooses its own delegates; it doesn’t matter who calls the convention. This is Agency law 101 and basic common sense.