Sorry, but I believe this analysis to be just wrong. The petite bourgesie lost the initiative in 1849, by the time a Brothers' War scenario happens the Prussian élites are the ones who matter the most and who were fairly reticent against stronger Catholic opposition; the Vatican isn't exactly helping matters either. Sure, unification as well masterminded by Bismarck is one thing, with more focus on the hated French than on internal division, but here there's no such scapegoat.
And while it's possible a minor or renegade Habsburg could have accepted the downgrade, the loss of legitimacy would be massive and a huge problem with other monarchs, both friendly and foreign, that saw Prussia essentially jeopardize ancient rights just because it suited them. There's zero chance Franz Joseph himself, or his immediate family, accepts to be "smoothly integrated by being demoted to König". Far from fixing problems, the two-Emperor conundrum would be a major source of internal strife by making the fight even more openly about who is the head honcho. Should a Kaiser submit to another? Neither wants to, and in this system, has plenty of reason and allies not to do it. After all, the King in Prussia elevated himself to fight against the wishes of the Kaiser, not in accordance with them.