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The way I understand it is the court will first need to determine if this is indeed a state v state matter, and if there are no other possible venues in which it can be heard. If there are other venues, the exclusive original jurisdiction would not apply, and the case would be denied.

Your understanding is not correct. SCOTUS has original jurisdiction over certain matters according to Article III. It doesn't need to accept all of its original jurisdiction. Thus, it can exercise discretion over how much of its original jurisdiction it exercises. It prefers not to hear many cases it could hear originally. It's Article III original jurisdiction does include any case in which a state shall be party. However, with state v. state cases SCOTUS by tradition does not defer jurisdiction to another court. In those cases, other non-state litigants, such as POTUS, may be granted leave to inteverne at SCOTUS.

Furthermore, there are legal arguments that can be made that SCOTUS could hear the case as original jurisdiction even if Texas is determined to lack standing. That gets into the technical relationship between the Declaratory Judgment Act, the Elveenth and Fourteenth Amendments. It is unprecedented, but possible. This case is about the legitimacy of presidential electors. POTUS and VPOTUS are not ordinary citizens with regard to their legal relationship with the several states. Something execptional is possible from SCOTUS.