weighted directed Walk in a graph
This problem is PP-hard under Turing reductions when your weights are encoded in binary. The lower bound follows by a reduction from the $K$-th largest subset problem. Given a set $S\subseteq \{ s_1,\ldots,s_n \}\subseteq \mathbb{N}$ and $B, K\in \mathbb{N}$, this problem asks whether
\begin{align*}
\# \{ S'\subseteq S : \sum_{s\in S'} s \le B \} \ge K.
\end{align*}
It has recently been shown that this problem is PP-complete under polynomial-time Turing reductions. It's easily seen how to encode this problem into yours (and see Proposition 4 in here in case you need some inspiration).

If $k_1$ is given in unary then your problem is PP-complete. If $k_1$ is given in binary then your problem is in PEXP, I cannot say if it is PEXP-complete.

In general, you often see this kind of problems in the setting of quantitative verification of (stochastic) systems.