I think that any scenario invoking thr 12th would quickly lead to a constitutional crisis, and almost certainly cause a convention to be called.
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.
Calling a constitutional convention, however, is another matter entirely. Two thirds of state legislatures would need to pass resolutions calling for a constitutional convention for that to actually happen. It's not too hard to imagine how this would play out. The Republican congress gives the Presidency to their man. Assume the Republican candidate is clearly not the winner of the popular vote. Maybe the backlash would force the Democratic Party to launch a campaign for a second constitutional convention (though I highly doubt either party really would want to open that Pandora's box), but the Democratic Party does not control nearly enough state legislatures to pull that off.
Maybe if the Republican legislature chose Trump (I) in a scenario where Cruz (R) placed fourth in the electoral college, you might see certain conservative states (like Utah) joining the Democrats in calling for a second constitutional convention, but even then I'm skeptical such a thing could be pulled off. There's too much inertia built into the system for a wholesale makeover of the US constitution to be accomplished "within the system."
