OK, I'm screwing up royally here.
Class 2 is going to be limited to 30V or less on the output, so there is no such thing as a
Class 2 transformer with a 120V output, sorry.
I was led astray by looking some catalog info and the fact that the 120V output transformers said they were UL listed under UL-5085-1, which I knew is the standard that covers Class 2. But I looked at the definitions themselves in the UL spec and Class 2 is limited to 30V on the output side. That UL spec does also include larger transformers with higher voltages, but they wouldn't be Class 2. A Class 3 transformer can be as high as 150V output, but the exceptions allowing for not having primary protection wouldn't apply, so it's a moot point.
Bottom line, you need protection on the primary as previously mentioned. If that breaker is 20A or less, you're good to go. If not, you will have to find somewhere to put one, it won't fit in that box with the starter.
http://www.functionaldevices.com/pdf/datasheets/TR50VA008.pdf