While only having access to the first edition, I'd say Hail Caesar can be just as (if not more) applicable to the first half of the Italian Wars considering its varied weapon types, troop types, and more in-depth melee combat. However, as the Italian Wars reached their later stage, armament and troop types became less varied, and battles in which one side didn't have a clear advantage over the other less so as well. Of course, most games/gamers seem to focus on the first half of the conflicts and there are very few suitable miniatures for the later period.
All of this is a very long-winded way to say that yes, I think Hail Caesar is just as good/better than Pike and Shotte provided you want less abstraction and are refighting the first half of the period.
Edit: note that one downside of Hail Caesar is that shooting is often applied to melee units. In the Italian Wars, shot could and would act independently of their melee formations as needed so maybe somehow finding a way to implement the Hedgehog rule from Pike and Shotte into Hail Caesar and removing/reducing the ranged attacks of melee infantry could make it more historically-accurate.