I'm not certain Boehner believes that Republican voters will abandon the party...just like GHW Bush didn't believe it. He thinks that stating his position - cut, cap and balance - then giving in to "the inevitable for the good of the country" will be enough to keep voters satisfied.
I think he may be miscalculating badly (something that isn't rare in politics, wherever they are practiced).
Dude you are right on the money. There is so much green at stake here it is no doubt that their influence will have a negative impact.