Labor Lawyers Advice?

Here, let me put it in it's simplest form. Only the Company can fire or discipline people.

What you assume is that the Union has power which it does not have.

Bingo.

I've heard repeatedly that unionized workers work for the union. That's completely wrong, an is a line of BS that has been propagated over the years.
The straight fact of the matter is, the employee works for, and is paid by the company. Further, it is the company that owns the employees job, not the union. The union doesn't pay the employee, it doesn't supply the job, the benefits, the place of employment, supply the job description. All of that is done by the company (employer).

On the other has, the union has no say in who works on a particular job, as long as the worker is paid the agreed scale for that position, and the employee has the proper training, or break in time for the position.

the union does not fire people, and does not recommend people to be fired. However there are other ways that a union can influence the termination of employ, of a union member.

For example: Little, or poor representation during a grievance hearing. All of us who have experience with unions have known or heard of an employee that the union didn't want around for one reason or another, and how they were fired for the smallest infraction. Well, when that happens, it's usually because the union reps weren't trying to save that person's job. The reason could be anything from having an argument with the Shop Steward to being a close friend of a supervisor, or not being one of the clique, or even no issue at all, the union boys just have someone who they prefer to have that position. I've seen it time and time again.

I've also seen deals made between the Union and the company to accomplish the same end. Yes, it does happen.

What S340 stated above is spot on... "Only the Company can fire or discipline people".

However, the union can (illegally) fail to give proper, or full representation if the employee in question is an undesirable to them. It, also happens.