No, you are not wrong. Class I towers (Amateur, CB) are exempt from the icing ratings according to TIA-222G. I had to take documents into my local B & P department showing this before they would let me proceed with my permit. It certainly doesn't hurt to have the higher rated tower but it was an extra expense you didn't need to pay.
John, I agree with you. It probably would have been unecessary for the added extra expense if I wanted to fight them on that issue. However, I felt lucky just to finally get the permits in hand.
In retrospect, I am so glad I purchased the heavier tower with the motor drive. The crank up with the motor drive is the only way to fly.
73, Jim AB4D