The setting pretty much assumes that, indeed, any covenant that finds itself on the path of a crusade would be in trouble; even non-Spring ones. However, I do not think the rules bear this out as-is. Magi are immensely powerful in the setting, and I believe they are strong enough so that a Summer covenant (let alone a Tribunal) could smash, confuse, or disperse any mundane army, even one enjoying the benefit of some people of True Faith, Divine tradition, or so on. The only way to avoid this is by more direct Divine intervention and/or extremely powerful Divine beings - such as Michael - and I think this is essentially the SG saying "Oh yeah?! Well, I'm God, and I'm Allpowerful, so THERE!", which isn't very satisfying.
I would advise against making this into an Order-vs-Church affair. Instead, have the other magi in the Tribunal largely lethargic about the whole military campaign. If it doesn't target their lands or raw vis, this makes sense. Perhaps have some Jerbiton magi working behind the scenes to ensure the covenants in the Tribunal are not affected.
There should be something about the PCs covenant that makes it take the brunt. Perhaps it has affiliated itself with the Cathars, for example, and the rest of the Tribunal sees the crusade as proper "karma" - perhaps even warning the covenant to be wary of interfering with the progress of the crusade, and sending Quaesitors to sniff around to make sure of that. Or maybe it isn't recognized yet, so the magi directing the crusade away from covenants don't feel like they need to protect it. Or maybe some mundane reason prevents them from diverting the crusade from this specific stretch of land, and the Tribunal decides it is better to just let the crusade wade through this single covenant if that buys everyone else peace and quiet.
Thus, while every other covenant is hiding itself by magic, is hidden in a regio, or is off the crusade's path - the PCs covenant alone finds itself in the warpath.
Do make the crusade more "Mythic" as much as possible. I would start very small, indeed with stories preceding and heralding the crusade itself. Have angels and divine miracle workers work against a local Cathar community, and see how the PCs respond. Have them drag a noble, and a small military contingent, into this affair, to see how they deal with armed mundane forces. This will give you a feel for the magic the PCs could wield and how they could defeat/trick the crusader armies.
When it comes to the crusade itself, crank up the Mythic elements. I would allow a "Moving Aura", simply by having mobile "shrines", relics held by the priests overseeing the crusade. Erect a large spherical aura equal to a Cathedral's around such a relic, and good aura around pivotal people in the army (such as the king). Have a weak Dominion aura present whenever the divine host of the crusade is, much like a village's fields. Have the papal legate leading the crusade wield Divine powers (via Powers and Methods), and have crusading orders built (at least at the level of sergeant and up) as Divine supernatural societies, too (again wielding various Methods and Powers, as well as initiations into other Mysteries and hence Virtues). Have the leaders of the crusade guarded by relics, which grant supernatural protection and magical effects as well as sheer Faith points. Have angels and other Mythic creatures/events, such as saints' intervention, occur 'regularly' in the course of the crusade. Also, have some magus or secret society aiding the crusade against the PCs for some reason. Perhaps a clandestine faction of Catholic magi secretly serving the Church, perhaps an overzealous Flambeau, perhaps a Tytalus using the mundane turmoil to foster his plans against the PCs...
I would also consider making the crusade Infernal, rather than Divine. This is a big choice to make. Note that an Infernal campaign will likely be accompanied by various deceiving demons, who act as 'angels', and that the magic wielded by the diabolists would be indistinguishable from Divine magic to Hermetic investigation for the most part.