Hi Kfolsom,
Unfortunately, no, there is no way to apply an additional container <div> with a Style Template. Style templates cannot really edit any Site Notice HTML except to add or remove some optional pieces, such as a logo or a close icon.
It sounds like what you are trying to accomplish is possible via the Style Template's CSS areas under the "Advanced" dropdown for the different components. Would you mind shooting me an email at jonah.thompson@crownpeak.com detailing your use case and style requirements for this scenario? Some screenshots would also be very helpful if you have some available for what the solution should look like.
Thanks,
Jonah