  To gain insight into the unconventional superconductivity of Fe-pnictides
with no electron pockets, we measure the thermal conductivity $\kappa$ and
penetration depth $\lambda$ in the heavily hole-doped regime of
Ba$_{1-x}$K$_x$Fe$_2$As$_2$. The residual thermal conductivity $(\kappa/T)_{T
\rightarrow 0\,{\rm K}}$ and $T$-dependence of $\lambda$ consistently indicate
the fully gapped superconductivity at $x=0.76$ and the (line) nodal
superconductivity at higher hole concentrations. The magnitudes of
$\frac{\kappa}{T}\cdot T_c|_{T \rightarrow 0\,{\rm K}}$ and
$\frac{d\lambda}{d(T/T_c)}$ at low temperatures, both of which are determined
by the properties of the low-energy excitations, exhibit a highly unusual
non-monotonic x-dependence. These results indicate a dramatic change of the
nodal characteristics in a narrow doping range, suggesting a doping crossover
of the gap function between the s-wave states with and without sign reversal
between $\Gamma$-centered hole pockets.
