Does the union have to take my case? Or can they refuse to pursue arbitration on my behalf?

No, a union does not have to pursue every case to arbitration.  Unions have a duty to fairly represent their members (called a "duty of fair representation"), but they do not have to pursue every case.  A rule of reason applies.  If the union acts reasonably, it may decline to pursue a bargaining unit member’s grievance if to do so would, in the union’s reasoned judgment, be detrimental to the collective well being of the bargaining unit in general.

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