I don't think you will be able to get what you want within a Tableau Dashboard. The reason is that even if you can move from one tab to another, the dashboard will eventually reload if a timeout is reached, and it will reset your timer along with it.
My suggestion is to create a webpage that contains the stopwatch you found and below the embebed dashboard in a specific iframe. If the dashboard reloads in this scenario, only the iframe where the dashboard is will be reloaded keeping the stopwatch moving on at the other part of the html.
I hope this helps.
Thank you for that suggestion! That works perfectly for what I need.