The biggest problem I see in all this will be the tit for tat exchange that goes along with anything in politics these days and this Convention will ultimately fail to produce anything.
First the whole balanced budget amendment will ultimately fail because Congress will never agree on what cuts should be made or which taxes will be raised. Even if it is a Federal law and Constitutional Amendment, how often has the Federal government (all three branches) gone around the Constitution these days anyway?
Second, what are the chances that 3/4s of the States ratify it if it means deep cuts in social programs, military, federal assistance, etc? Because first you have to have Congress get on board with a balanced budget which it has been unable to do except sporadically through the last 40 years. And further, the people have to accept the fact that government will have to cut significant programs to get the books balanced. And that's something that's just not going to happen with either side of the political equation as news agencies will hype up the cuts as detrimental to whatever fundamental doctrine they hold dear.
And last, but not least, LG hit on the subject (although in a typical partisan fashion) of political agendas getting involved in this. Far be it while we are looking at the Constitutional Amendment process for someone to get the idea to toss in their own agenda and why not revise that pesky 2nd Amendment to reflect "modern day attitudes" with clear modern language that starts putting limitations on the type of firearms allowed? Or add in an Amendment concerning religion in schools?
And the Convention will break down along those partisan lines. Or start the you give me this and I'll give you that type of bickering. You get a revision of the 17th Amendment if I get a revision of the 2nd Amendment. And will break down without anything getting accomplished. Or will include changes that the people won't accept for the 3/4s majority.