= Solution
Multiply each time-sliced path weight by $\exp[-\sum_jV(x_j)\Delta t]$. A nonnegative finite $V$ is a local killing rate: the Feynman-Kac weight attenuates paths in proportion to the <time> spent in regions where $V>0$. Let that rate tend to infinity outside the allowed interval. Paths that exit are removed, leaving <Brownian motion> killed at its first boundary hit.
Consequently the required kernel is the Gaussian path measure restricted to surviving paths, equivalently
$$
\boxed{K_{|x|<L}
=\int_{x_i}^{x_f}\mathcal D x\,
e^{-\int[\dot x^2/(4D)+V(x)]dt},}
$$
with the infinite-barrier interpretation stated above. It solves $\partial_tK=D\partial_x^2K$ in the interior with absorbing <Dirichlet boundary conditions> at $\pm L$. A <reflecting boundary condition for a diffusion> would describe a different process.
Strictly, this is a <transition probability> density in the final <position>, jointly with survival; a <probability> for an interval of final <positions> includes an integral over $x_f$. Its integral over the whole allowed interval is at most one. It is not the normalized endpoint density conditional on survival, which would require division by the <survival probability>. In these <diffusion> units $V$ and the <eigenvalues> of $H$ have units of inverse <time>.
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