  This paper considers two logics. The first one,
$\mathbf{K}\mathsf{G}_\mathsf{inv}$, is an expansion of the G\"odel modal logic
$\mathbf{K}\mathsf{G}$ with the involutive negation $\sim_\mathsf{i}$ defined
as $v({\sim_\mathsf{i}}\phi,w)=1-v(\phi,w)$. The second one,
$\mathbf{K}\mathsf{G}_\mathsf{bl}$, is the expansion of
$\mathbf{K}\mathsf{G}_\mathsf{inv}$ with the bi-lattice connectives and
modalities. We explore their semantical properties w.r.t. the standard
semantics on $[0,1]$-valued Kripke frames and define a unified tableaux
calculus that allows for the explicit countermodel construction. For this, we
use an alternative semantics with the finite model property. Using the tableaux
calculus, we construct a decision algorithm and show that satisfiability and
validity in $\mathbf{K}\mathsf{G}_\mathsf{inv}$ and
$\mathbf{K}\mathsf{G}_\mathsf{bl}$ are PSpace-complete.
