Please be more specific...IF you got a ticket, post the violation section, THEN better answers will follow.
Your SD drivers license allows you to drive in any of the 50 States. It does not allow you to violate any safety laws or regulations. In answer to your question, no you wouldn't need the air brake endorsement although I think the knowledge is beneficial.
As far as your livestock trailer being commercial, then you'll have to tell us the trailer length, what you were hauling and if you had all the required safety equipment. I'm guessing you didn't simply get pulled over because you were tooling along with a single horse trailer at a safe 50mph, minding your own business. Sounds like you stepped in some doodoo and received an attitude adjustment ticket......just sayin' we need more specific information.....Dennis