I do agree that his military mind plays largely at it (that was always my answer to those "but why doesn't he challenge Meredith??" arguments - uhm, he's military and she's his commanding officer, it doesn't work like that), but I also see a degree of manipulation to it, depending on the argument or the method. Cullen himself admits that Meredith manipulated him with some level of gaslighting, and some of those war table missions when he really really doesn't wanna do something and then sounds just as disgusted after doing it as you'd expect just... rub me the wrong way (the one with Florianne's head in particular really makes me uncomfortable, I don't know).
There's also that line he says in the Champions quest, about it being "his turn" to question and "letting you" turn the Inquisition astray... it sounds like he let you get away with a lot, watching others have their "turn" and not stopping you, not questioning what "you" were doing, and he regrets that on a more personal level than just following orders (as much as he does with Meredith). It feels as more than just subservience, you know? Feels more like he was again easily manipulated by an authority figure preying on his disciplined nature of obeying higher commands. So there's military mind at work, sure, but there's also some emotional manipulation involved, depending on how you read it.
I think you're probably right. I think there are also some differences in his relationship to the IQ and the Inquisition compared to Meredith, in that the way he was recruited and the role he was asked to play is also different. While I still think that he was once very much committed to the Templar Order's principles (and that that's why he ultimately overthrew Meredith), he was still primarily a cog in a much bigger wheel. With the Inquisition, in contrast, he was brought in at the top to help run it and shape what it would become. So although he still runs the Inquisition forces in a military style, I feel like his investment in it is a bit more personal as a result, because what it becomes is also partially his fault. So, that I could see him taking very personally the Inquisition's failures, like in those Champions of the Just scenes.
It then makes me wonder how he would go about challenging an Inquisitor who had gone off the rails. Would it be similar to Meredith and he'd ultimately have to overthrow you? Or, does the evil!Inquisitor still listen to Cullen enough that he'd feel he still had a chance to help redirect the organization? Or, would there be a point where he would ultimately even walking away? This is of course all further muddied if he happens to be in love with the Inquisitor. But even without that, it would be an interesting question.
I can't help but think that it would be even harder to see the Inquisition go down in flames since in this case, he would be partly responsible. Part of me thinks he might fight harder for it than he did for the Order, while part of me wonders if he'd just get... tired at some point, after two failed organizations, and finally walk way and start to just take care of himself instead of the rest of the world.