Large-language-model (LLM) graders promise to relieve the grading burden of upper-division STEM courses, but most deployments to date send student work to third-party APIs, violating FERPA and exposing institutions to data risk while requiring substantial assignment modification. We present $\textbf{LaTA}\ (\textit{LaTeX Teaching Assistant})$, a drop-in, open-source autograder that runs entirely on commodity on-premises hardware and assumes a LaTeX-native workflow already adopted by many engineering and physics courses. LaTA implements a four-stage pipeline (ingest, segment, grade, report) using a locally hosted open-weight chain-of-thought LLM grader (gpt-oss:120b) that compares student work to an instructor-authored reference solution and applies a YAML rubric with binary per-item scoring. We deployed LaTA in Winter~2026 in ME 373 (Mechanical Engineering Methods) at Oregon State University, grading every weekly assignment for approximately 200 students on a single Mac Studio at \$0 marginal cost per assignment and 1--3 minutes of wall-clock time per submission, enabling regrading of corrected assignments and greatly expanded TA office hour offerings. The instructor-confirmed grading-error rate held at roughly $0.02$--$0.04\%$ per rubric line item across the term. Relative to the same instructor's previous traditionally-graded cohort, the LaTA-graded cohort outperformed by approximately $11\%$ on the midterm exam and $8\%$ on the final exam, and reported large gains in self-assessed confidence on every stated learning objective ($N = 159$ survey responses, $Δ\geq +1.49$ Likert points, $p < 10^{-27}$ on every comparison). We release the code under AGPLv3.