Doesn't them being Royalists mean they were loyal to the Therin line? Anora is just Cailan's wife, from that point of view and they'd much more readily look to Alistair if his royal lineage was ever revealed.
Anora is still the queen, even being referred to as such consistently throughout the game. As such, she's technically a royal
As for whether Bryce would be loyal to solely the Theirins or not, it's not overtly stated what their branch of royalism is, only that they fought against Calenhad at one point but swore an oath to him upon being beaten (which doesn't mean they're like Eamon in that they believe a Theirin must always rule).
But Bryce is still a shrewd guy. We know the Couslands are ardent royalists, and while that can mean supporting a particular dynastic claim like the Theirins, that to me speaks more of traditionalism. In the case of Ferelden, this would tie into a sentiment that the Theirin bloodline is Ferelden.
Me, I happen to believe that a man like Bryce, who condemns Howe for trying to use the chaos to advance himself, would stick with the throne's sole remaining monarch -- particularly since at that time, no one knew Alistair or the Warden survived, so if he knew about Alistair's existence there wouldn't be much he could do -- in an effort to prevent the realm from falling apart even further, more so since he would've been at Ostagar and seen how bad it was.
Assuming the battle would've gone more or less the same and that Bryce wouldn't have been able to bring Cailan under control. If Cailan had been brought under control the battle plan would've been different, as the battle plan Loghain devised was one he had to devise because Calian "wanted a battle the bards would sing of for centuries" and where a "king rides with the fabled Grey Wardens". Loghain had to make a battle plan that would satisfy Cailan's ego and lust for glory while making the best use of the information at hand and their dwindling manpower -- Note: Ostagar's structure is better served with the soldiers not on the outside.
If Bryce had returned to Denerim, he would've served as a calming sentiment among the Bannorn. Two Teyrns standing with Queen Anora to prevent Civil War, while corroborating each other, and Bryce serving as perhaps a chancellor to the throne (I imagine Bryce would've convinced Loghain that he need only be the Commander of the Army and Anora and he (Bryce) could handle the politics).
I think that it's either in Why Teryn Loghain is the deepest character in Dragon Age, or The Complete Defense of Loghain Mac Tir. I did succeed in finding the latter at one point recently, but can't now. The former is over 500 pages and if there's a way to just find posts by devs, I don't know how.
I managed to cut in the tack-on that allows for dev posts to be seen for the former, and none exist. I've seen the dev posts in the latter before and unfortunately it wasn't in that one (and also, some of Gaider's posts there make the timeline even worse then it already was from just in-game material).