I'm still in favour of Case Eagle, mostly for political reasons rather than the military ones already outlined. Our pretext for this war only involves Austria, not our other enemies. If we take a defensive posture on the other two fronts, then it looks a little bit less like we're making a play to become hegemon of Europe, and a little bit more like we're just trying to unite the German nation once and for all.
It almost worked in the leadup to World War 2, and it might be helpful now. Some decision-makers in other countries will be inclined to let us go that far if we can be stopped from going any further, which will affect their calculations in a way that works in our favour.