Unfortunately, this may not be possible.
After activating the JPEGmini plugin, are all images exported by Lightroom (whether from the Library module or the Web module, where the Juicebox Plugin for Lightroom is) processed by the JPEGmini plugin?
Have you tried exporting a Juicebox gallery with the JPEGmini plugin both enables and disabled to see if there is any difference in the resulting image size? (I do not have JPEGmini so cannot run such a test myself.)
If not, then it sounds like JPEGmini might need to provide an API to allow developers of other Lightroom extensions to utilise their plugin's functionality (and, from taking a look at their website, there does not seem to be one).
Even if it was possible to hook into the JPEGmini functionality, the Juicebox Plugin for Lightroom would need to be rewritten to incorporate such functionality (the user interface and underlying functionality of the Juicebox Plugin for Lightroom is not something that can be modified by users as the web engine is compiled and cannot be edited) and I think it is unlikely that the developers would be willing to rewrite the Juicebox Plugin for Lightroom to support a non-free third-party plugin.
If the images exported for a Juicebox gallery are not automatically processed by JPEGmini when the JPEGmini plugin is active (i.e. if Lightroom's exporting functionality is not automatically intercepted and overridden by JPEGmini in all scenarios), then it sounds like your current workaround is, indeed, the best course of action.
I realise that this is not a solution to your problem but I hope that it at least helps to clarify things.