We present a multi-agent, human-in-the-loop workflow that co-designs quantum error-correcting codes with pre-scribed transversal diagonal gates. It builds on the Subset-Sum Linear Programming (SSLP) framework, which partitions basis strings by modular residues and enforces$\mathbf{Z}$-marginal Knill-Laflamme (KL) equalities via small LPs. The workflow is powered by GPT-5 and implemented within TeXRA, a multi-agent research assistant platform where agents collaborate in a shared LATEX-Python workspace synchronized with Git/Overleaf. Three specialized agents formulate constraints, sweep and screen candidate codes, exactify numerical solutions into rationals, and independently audit all KL equalities and induced logical actions. Focusing on distance-two codes with nondegenerate residues, we catalogue new nonadditive codes for dimensions$K \in\{2,3,4\}$on up to six qubits, including high-order diagonal transversals, yielding 14, 116 new codes. From these data, the system abstracts closed-form families and constructs a residue-degenerate ((6, 4, 2)) code implementing a transversal controlled-phase$\operatorname{diag}(1,1,1, i)$, illustrating how AI orchestration can drive rigorous, scalable code discovery.