There is a report setting under option, even same setting is also there in Power BI service - Hide the visual header under visual option (Power BI desktop) Under setting visual options ( Hide the visual header in reading view) - In Power BI service
Or If you make this change , visual header will not be available for any visual in Power BI adn user will not able to use export functioanlity in Power BI.
You can also use Export data "none"
This can be one of the fix for your curretnt situation. BY doing so, user will not abe able to export data from report.