The House of Representatives has actually passed two debt ceiling plans and sent them to the Senate. Both were killed on arrival because they contained a requirement that Congress pass a balanced budget amendment and send it to the states. Even some Republicans voted against the idea, not because they oppose the amendment, but because they've resigned themselves to the idea that the Democrats wont go for it.
Meanwhile, back at the ranch, 70+% of Americans polled are in favor of a balanced budget amendment. Why are supporters in Congress not taking this case to the people instead of bickering with other lawmakers and media pundits?
It seems to me a good strategy would be to pass a debt ceiling hike with the provision that a vote be taken in both houses on the balanced budget amendment within 60 days. Take those 60 days to appeal to the public to put pressure on their representatives to pass the balanced budget amendment. If public sentiment is where the polls say it is, Democrats will come around. If it's not, so be it.
In any case, helping to kill your own proposals because you assume you're going to lose before you even start the fight is weak. Give yourself time to make your best pitch to the masses. If they aren't moved, you did all you can do.