Montfort objected to the 'redirecting' of the fourth crusade because it was against Pope Innocent III's direction - I think this is testimony in fact to the sincerity of the beliefs of at least certain crusading nobility at the time. Innocent III, after railing at the crusaders for invading and sacking Constantinople, 'accepted' the fact once it became fait accompli.

Simply put, at 1203 the Eastern Church was still in schism with the western Church, but not declared heretics by the Roman Catholic Papacy. That was a pretty significant difference theologically and legally (under Canon law, that is, as well as incorporations of heresy in the civil law of some monarchs like Barbarossa). By the time Simon de Montfort descended into the territories of Count Raymond of Toulouse and the Viscount Raymond-Roger of Beziers and Carcassonne (1209), these southern nobles were under interdict and excommunication with their lands forfeit (as declared by the Papacy). There really was nothing for the crusading northern nobility to justify - the dirty theological work had already been done.