Core idea - Count Range Instead of Exact

In Fixed-Length Paths I, you counted pairs with $d_1 + d_2 = k$.

Now you count pairs with $k_1 \le d_1 + d_2 \le k_2$. The structure is identical, only the counting changes. For each distance $d_1$, you want distances $d_2$ where $k_1 - d_1 \le d_2 \le k_2 - d_1$. Use a frequency map or sorted list to count matches in that range. The rest of the algorithm is identical: find centroid, count through it, subtract same-subtree paths, recurse. Only the pair-counting logic changes.