Don't have a list of Texas recognized lodges handy, but know for a fact that the blue GL of Minnesota is recognized as legitimate by the GLoT. Given the current limited recognition of the PHGLoT, however, would guess that the PHGLoM is not.
Is it a blue GL or PHA GL? If blue, Bro. Cook will (or at least SHOULD) be fully welcomed and treated as any other brother. If PHA, he will (or again, SHOULD) be treated as a respected visitor and personally as a brother, however, would not be allowed to attend closed functions or engaged in "masonic" conversation. Your membership in either lodge would be irrelevant with respect to the issue. Minnesota rules apply to you ONLY if you're within the state.
At least, that's my interpretation based on the current state of things.