as salam alaikum
the four Sunni schools of jurisprudence are related to practical laws and therefore they are not directly linked to doctrinal and theological issues (although sometimes we see some affiliations).
Theologically, Sunni schools can be divided in three main branches: Ash'aris, Maturidis and Ahl al-Hadith.
The Sunni-Shia debate have been dealt in different way by different scholars belonging to these branches. Therefore the judgment more often goes back to ulama, or groups of ulama, and not to the schools themselves. Also, it may depends on what Shi'i trend they refer as their judgments may differs from a Shi'i approach or group to another.
We certainly see some Sunnis very staunch against Shi'ism while others are more tolerant. The same goes for the Shias towards Sunnis. So it is prudent not to generalize and analyze single groups and opinions from case to case.
With prayers for your success.