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Yes, he can. The "residency" requirement for a court to have jurisdiction over the military member is not the same thing as the concept of your legal residency for tax purposes. So, the state that you claim does bar you from also meeting the "residency" requirements to file in other states. You just have to be present in the state for the requisite period of time.
Furthermore, the fact that the mother and child also live there clearly establish that state's jurisdiction over the question.