Given a complete, weighted and undirected graph $G$, complexity of finding a path with a specific cost
Given an instance $\langle \{s_1,\ldots,s_n\}, T \rangle$ of subset sum, construct the following weighted graph. The vertices are
$$ v_0,v_1,\ldots,v_n,u_1,\ldots,u_n. $$
Connect $v_{i-1}$ and $v_i$ with an edge of weight $s_i$. Connect $v_{i-1}$ to $v_i$ via $u_i$ (i.e., add the edges $\{v_{i-1},u_i\},\{u_i,v_i\}$) with zero weight edges. There is a simple path of total cost $T$ iff there is a subset of $\{s_1,\ldots,s_n\}$ summing to $T$. This shows that your problem is NP-hard, and in fact NP-complete, since it's clearly in NP.