The union is there to enforce the collective bargaining agreement between the employer and the employees. It is their duty to enforce the terms and conditions of that agreement even if the result will harm one of their members. The union looks out for what is best for the membership as a whole, not based on the individual.
The current process to sue a union would be as an unfair labor practice charge against the union. The charge is commonly known as "Failure to Represent". All unfair labor practice charges under the National Labor Relations Act have a time limit to file. The union has no special protection. If they intentionally act against you without a bonafide reason to do so, they are subject to an unfair labor practice charge.
It is not difficult to sue the union, but it is difficult to be successful, just as it is in most law suits. Anyone can sue, but to win the lawsuit is often difficult.