If the House follows the procedure for selecting the President outlined in the 12th amendment then there would not be a constitutional crisis. There is no ambiguity in how that would play out; we have a historical example from 1824. Of course a 21st century Corrupt Bargain would be hugely unpopular; the popular backlash would certainly dwarf the combined strength of the Tea Party and Occupy movements.
For decades they said that the popular vote winner would never lose the electoral vote, and if they did, the popular outcry would be enormous and the electoral college would be thrown out. 2000 happened, and we still have the electoral college.