Generators for K-theoretic Hall algebras of quivers with potential

K-theoretic Hall algebras (KHAs) of quivers with potential $(Q,W)$ are a generalization of preprojective KHAs of quivers, which are conjecturally positive parts of the Okounkov-Smironov quantum affine algebras. In particular, preprojective KHAs are expected to satisfy a Poincar\'e-Birkhoff-Witt theorem. We construct semi-orthogonal decompositions of categorical Hall algebras using techniques developed by Halpern-Leistner, Ballard-Favero-Katzarkov, and \v{S}penko-Van den Bergh. For a quotient of $\text{KHA}(Q,W)_{\mathbb{Q}}$, we refine these decompositions and prove a PBW-type theorem for it. The spaces of generators of $\text{KHA}(Q,0)_{\mathbb{Q}}$ are given by (a version of) intersection K-theory of coarse moduli spaces of representations of $Q$.