I'm not sure you could create a "button" that would do this in Desktop. I suspect that you could possibly do this with Flow triggered by a Power BI refresh but your best bet is probably to use Power App perhaps combined with Flow but I'd need to know more about your setup and source systems and how Power BI would fit with that.
There is a Flow template out there, "Trigger a flow with a Power BI data driven alert". There are also flow templates for sending emails. So the question in that case would be can you send the list of guilty members to the Flow or access that list from Flow.
There is another Flow for "Send an email to any audience when a Power BI data alert is triggered".
So, in the Flow world, your interaction is going to be around Power BI data alerts. So, if you want to do on-demand, you'll probably have to look into Power Apps. The only problem is, there is no data connection for Power BI in Power Apps right now. So, you'd have to access the source data systems where you are getting your information from for your Power BI Desktop file.
