Let me get this right.
According to the mechanic, your brother, acting as your agent, authorized work on the trailer for $900. When signing the paperwork, he may have gotten the name wrong. As your agent, he wrote the check to pay the bill.
According to you, when you returned, you discovered all these things your brother had done in your absence and were horrified. You cancelled the payment on every check he signed arguing that he was not authorized to sign anything so you don't owe. You are using the fact the paperwork is in the wrong name and that you operate as a sole properitor as proof that your brother was NOT acting as your agent.
However, your story has holes in it. Clearly you do have a relationship with your brother. You called him an employee in a letter to their attorney. You are now scrambling to take back that statement because you realize that it makes him sound like he could have been your authorized agent in this situation. I'm guessing that out of all the bills he paid, this is the only one you are failing to pay....
Whether your brother is an employee or contractor is probably not going to change the outcome of this lawsuit. An employee is someone you are supposed to report a W-2 on, pay workman's comp, control their work, etc. A contractor gets a 1099-Misc. That usually matters for tax purposes and with the labor board.