My answer to whether you have a case: maybe.
I agree with the above answer that you have an evidence problem with proving your damages. Since you no longer have access to the SUV, it will be difficult for you to show that the transmission could not be opened. You could try to present witness testimony from the two shops, but that might be an uphill climb.
However, depending on the language of the contract with the warranty company you may have an action to bring under the Michigan Consumer Protection Act (MCPA), or a fraud or breach of warranty claim. At the very least, you can file a complaint against the warranty company to the Michigan Attorney General.
I agree with the first answer, that you may hire an attorney at any time. More specifically, you may be asking 1) whether your union has guidelines for its members about this, or 2) when it would be most beneficial to hire an attorney. Without seeing the bargained agreement for your unit, I cannot say about any guidelines or restrictions. But as to when it is beneficial, an attorney could help in many ways throughout your internal appeals process - at a minimum, one could even just assist you and your union reps identify legal issues and formulate better arguments.
Ultimately, you will only know if you contact an employment and labor attorney to sort through the issues of your situation in detail.
In order to answer fully, I would need more facts. But regarding joint and several liability, you are correct that it means the plaintiff suing your employer is seeking recover all the damages from any of the defendants regardless of their individual share of the liability. The general rule for liability in Michigan, however, is that a defendant is severally liable for damages attributed to its percentage of fault. There are several exceptions to this rule, though, which could lead to the application of joint and several liability.
From the information you give, it also sounds as though the legal theory of "agency" is being brought into question. In other words: Were you in your capacity as an employee or independent contractor acting as an agent for the company? And also, were you acting within the scope of your employment when dealing with this customer? There are many other factors to consider, but... If you were an agent, and were acting within the scope of your employment, your chances of being held personally liable are less.
Since you are personally named, I strongly recommend that you hire an attorney to help you sift through all of the facts and determine a legal strategy to properly defend yourself.