I use Brave too, go to Settings (top right Menu button) scroll down till you find Privacy and Security, and you find the Cookies & Other Site Data section there.

If you scroll down you'll see 2 important sections, Clear cookies when site is closed and Never allow cookies. To get rid of existing cookies, add Twitter to the first, open the window/tab the close the site. It should hopefully have removed any cookies when you close the window or tab (also open the mobile.twitter,com link too).

Then add Twitter to the Never Allow cookies section and hopefully you should be good.

If it doesn't work and you still can't access Twitter, you could try clearing all cookies in the main Clear All Browsing Data. Close the browser, (but that would mean you have to login everywhere again), make sure that Twitter is set in the Never Allow cookies section (and the mobile.twitter.com url too) then open up Twitter again. If you see that login, install the extension and bobs yer uncle.

Are you using a password generator? That might be logging you in (I use Last Password) if so, remove your Twitter logins from that site if you can't switch off the auto login attempt. Or see if you can remove from browser storage, don't know where they would be stored though.

Hope you understood that, I'm crap at explaining things! ;-)