Voting on Taxes

Apparently all the Blue Dogs in the House are terrified of being asked to vote on a bill that would extend only the tax cuts on the middle class. Jonathan Chait comments:

So the issue here is that they're afraid a vote to extend tax cuts will be turned into a vote to raise taxes, and thereby into a vote to raise taxes on the middle class. Okay, I kind of get that — this presumes massive communicatory incompetence by these Democrats, but that may be a fair assumption. So why not just hold two different votes? They can vote for both the universal tax cuts and the upper-class-only tax cuts. If both bills pass, Obama can sign the first and veto the second. If Republicans block the universal tax cuts, Democrats can make that their campaign issue.

Or, better yet, a vote on the middle-class-only cuts followed by a vote on the upper-only cuts. Either way, though, I assume this is too easy a solution and doesn't work because there's not enough time to schedule two votes? Or the Blue Dogs don't want two votes? Or something. Not sure what, though.