Service worker cache super-cedes browser cache and honestly you should focus more on it than browser cache.
There is not a way to purge browser cache via the service worker or UI script for that matter. You have no control over that.
Even setting the Cache-Control header may not really matter. Browsers can be more aggressive as they deem necessary. Routers, load balancers, etc call get in the way as well.
If you are using service worker cache you can invalidate it any way you want. I have made some very sophisticated service workers, especially in recent months around invalidating cached assets in SW cache and indexedDB.
It is way more complex than I can share here LOL. There are numerous strategies you can employ, it all depends on what goals you have and what call the persona of your network addressable assets :)