  In arXiv:1911.08172 we have studied the single-particle free energy of a
class of Little String Theories of A-type, which are engineered by $N$ parallel
M5-branes on a circle. To leading instanton order (from the perspective of the
low energy $U(N)$ gauge theory) and partially also to higher order, a
decomposition was observed, which resembles a Feynman diagrammatic expansion:
external states are given by expansion coefficients of the $N=1$ BPS free
energy and a quasi-Jacobi form that governs the BPS-counting of an M5-brane
coupling to two M2-branes. The effective coupling functions were written as
infinite series and similarities to modular graph functions were remarked. In
the current work we continue and extend this study: Working with the full
non-perturbative BPS free energy, we analyse in detail the cases $N=2,3$ and
$4$. We argue that in these cases to leading instanton order all coupling
functions can be written as a simple combination of two-point functions of a
single free scalar field on the torus. We provide closed form expressions,
which we conjecture to hold for generic $N$. To higher instanton order, we
observe that a decomposition of the free energy in terms of higher point
functions with the same external states is still possible but a priori not
unique. We nevertheless provide evidence that tentative coupling functions are
still combinations of scalar Greens functions, which are decorated with
derivatives or multiplied with holomorphic Eisenstein series. We interpret
these decorations as corrections of the leading order effective couplings and
in particular link the latter to dihedral graph functions with bivalent
vertices, which suggests an interpretation in terms of disconnected graphs.
