  We study the Feynman graph structure and compute certain exact four-point
correlation functions in chiral CFT$_4$ proposed by \"{O}.G\"{u}rdo\u{g}an and
one of the authors as a double scaling limit of $\gamma$-deformed
$\mathcal{N}=4$ SYM theory. We give full description of bulk behavior of large
Feynman graphs: it shows a generalized "dynamical fishnet" structure, with a
dynamical exchange of bosonic and Yukawa couplings. We compute certain
four-point correlators in the full chiral CFT$_4$, generalizing recent results
for a particular one-coupling version of this theory -- the bi-scalar "fishnet"
CFT. We sum up exactly the corresponding Feynman diagrams, including both
bosonic and fermionic loops, by Bethe-Salpeter method. This provides explicit
OPE data for various twist-2 operators with spin, showing a rich analytic
structure, both in coordinate and coupling spaces.
