On Chrome, theres an option to list cookies for a site by clicking on that padlock to left of site url, so quicker way of clearing cookies for your Jenkins site. Maybe FF has similar setup.

Then hard refresh browser should clear cache I think (ctr/cmd + shift + r on any system or ctr + f5 on windows).

It doesn't clear cache but it will automatically delete any cookies when you close a tab or when you change the domain. That might do what you need, when you use it in conjunction with Containers.

If you're okay with using chrome: create another profile and only use it for jenkins, in fact, set up a shortcut that opens it in app/kiosk mode and maybe see if there's a commandline flag to clear all history every time you open it, that way all you need to do is click the shortcut and everything else happens in the background and it doesn't affect the rest of your browsing exp.