Mizuno's rank three Nahm sums II: identities of index $(1,2,2)$ and modular forms

Mizuno provided 15 examples of generalized rank three Nahm sums with symmetrizer diag(1,2,2) which are conjecturally modular. Using the theory of Bailey pairs and some q-series techniques, we establish a number of triple sum Rogers--Ramanujan type identities. These identities confirm the modularity of all of Mizuno's examples except for two non-modular cases. We show that the two exceptional cases of Nahm sums are sums of modular forms of weights $0$ and $1$. We also prove Mizuno's conjectural modular transformation formulas for two vector-valued functions consisting of Nahm sums with symmetrizers diag(1,1,2) and diag(1,2,2).