The tree edit distance is a natural dissimilarity measure between rooted ordered trees whose nodes are labeled over an alphabet $Σ$. It is defined as the minimum number of node edits (insertions, deletions, and relabelings) required to transform one tree into the other. In the weighted variant, the edits have associated costs (depending on the involved node labels) normalized so that each cost is at least one, and the goal is to minimize the total cost of edits. The unweighted tree edit distance between two trees of total size $n$ can be computed in $O(n^{2.6857})$ time; in contrast, determining the weighted tree edit distance is fine-grained equivalent to the All-Pairs Shortest Paths problem and requires $n^3/2^{Ω(\sqrt{\log n})}$ time [Nogler et al.; STOC'25]. These super-quadratic running times are unattractive for large but very similar trees, which motivates the bounded version of the problem, where the runtime is parameterized by the computed distance $k$, potentially yielding faster algorithms for $k\ll n$. Previous best algorithms for the bounded unweighted setting run in $O(nk^2\log n)$ time [Akmal & Jin; ICALP'21] and $O(n + k^7\log k)$ time [Das et al.; STOC'23]. For the weighted variant, the only known running time has been $O(n + k^{15})$. We present an $O(n + k^6\log k)$-time algorithm for computing the bounded tree edit distance in both the weighted and unweighted settings. Our approach begins with an alternative $O(nk^2\log n)$-time algorithm that handles weights and is significantly easier to analyze than the existing counterpart. We then introduce a novel optimization that leverages periodic structures within the input trees. To utilize it, we modify the $O(k^5)$-size $O(n)$-time universal kernel, the central component of the prior $O(n + k^{O(1)})$-time algorithms, so that it produces instances containing these periodic structures.