Combinatorial Testing (CT) is essential for detecting interaction-triggered faults, yet generating minimal Covering Arrays under complex constraints remains an unresolved NP-hard challenge. Current greedy algorithms are highly scalable but suffer from severe ``diminishing returns'': they efficiently cover initial interactions but produce bloated, redundant test suites when struggling to pack the final few difficult pairs. While exact mathematical programming could theoretically address this inefficiency, it has historically been intractable due to combinatorial explosion. In this paper, we pioneer the application of exact mathematical modeling to CT by introducing SeqTG, a scalable framework based on Sequential Integer Linear Programming (ILP). To circumvent the scalability barrier, SeqTG employs a novel Warm-Start strategy: a rapid greedy initialization first clears the ``easy'' interactions, allowing the rigorous ILP solver to exclusively optimize the fragmented, difficult-to-cover remainder. The pipeline operates in three stages: (1) a Constraint-First phase grouping must-include requirements via graph partitioning; (2) an Incremental Optimization phase targeting the remaining interactions with sequential ILP; and (3) a Global Minimization phase eliminating redundancies via set-covering. Extensive evaluations across standard benchmarks and 200 large-scale configurations validate the framework's efficacy. The results demonstrate that SeqTG effectively eradicates late-stage bloat, achieving state-of-the-art test suite compactness and strict constraint adherence.