  The relative intensity distribution in the rotationally resolved
laser-induced fluorescence spectra belonging to the $E(4)^1\Sigma^+\to
(a^3\Sigma^+_1,X^1\Sigma^+)$ band systems of the KCs molecule was analyzed. The
experimental intensities in doublet $P$,$R$ progressions assigned to
spin-allowed $E\to X$ and spin-forbidden $E\to a$ transitions up to their
common ground dissociation limit were described in the framework of a
coupled-channels (CC) deperturbation model applied for the interacting \Xstate\
and \astate\ states. The CC intensity simulation was based solely on fixed
electronic structure parameters as functions of the internuclear distance $R$,
namely: accurate empirical potential energy curves for all three states,
\emph{ab initio} estimates for matrix elements $A(R)$ of the hyperfine
structure (HFS) , and transition dipole moments $d_{EX}(R)$ and $d_{Ea}(R)$. A
comparison between the measured intensities and their theoretical counterparts
demonstrates a strong competition between different intramolecular
interactions. A weak spin-orbit coupling of the upper \Estate\ state with the
remote $^3\Pi$ states is responsible for appearance of the $E\to a$ vibrational
bands for the intermediate $v_a$-values. In turn, the HFS coupling between
\Xstate\ and \astate\ states leads to peculiarities in $E\to (X,a)$
intensities, which are pronounced for high $v_{X/a}$-values in the vicinity of
K(4$^2S$)+Cs(6$^2S$) dissociation threshold. Both adiabatic ro-vibrational and
non-adiabatic electronic-rotational interactions explain the abrupt deviation
of some observed $P/R$ intensity ratios from the expected H\"{o}nl-London
factors.
