Conversely, the House could start passing bills that fund everything but certain parts of ObolaCare. The Senate could pass them and Obama could veto them (shut down the government).
Completely agree!
Try to get the whole thing killed at once and when that fails take it apart piece by piece. Make obama veto everything and make the Rats support his veto. I don't know if you even need to shut down the govt and it might be counterproductive, because the media will twist who's fault it is and obama will never agree to repeal. However, if you keep passing bills that are popular and that cut the legs out of obamacare the support for Pubs will grow.
I could see Cruz, Lee and some other Pubs doing this but not McConnell.