Without seeing a packaged workbook to play with the data, it's going to be tough for us to give you a helpful answer. I'm thinking that you may be able to create a dashboard with multiple sheets then use one sheet to filter on the other, ensuring you always have something showing on the viz and it never actually empties.
Did you try using the show empty rows and show empty columns from the Analysis Menu ?
Hi to all and thanks for your answers.
In the meantime a colleague helped me in order not to have a "None" message in the final dashboard when an empty intersection happens: we used the same answer of Marc Moussa. He even instructed me on how to generate a packaged workbook.
Then my boss told me that she would have accepted the empty result, so there is not this necessity anymore. For the moment!