Hi, thane. The absolute bare minimum configuration is this:
$HOME/.config/BraveSoftware/Brave-Browser/First Run
$HOME/.config/BraveSoftware/Brave-Browser/Local State
$HOME/.config/BraveSoftware/Brave-Browser/Default/Bookmarks
$HOME/.config/BraveSoftware/Brave-Browser/Default/Preferences
Perhaps you could create such a barebones BraveSoftware directory, rename it BraveSoftware-minimum, then create a wrapper script which you always use to launch the browser. The wrapper script can be anywhere in your PATH that takes precedence over /usr/local/bin (e.g., $HOME/tc/.local/bin/brave-browser) and can look like this:
#!/bin/sh
cp -r $HOME/.config/BraveSoftware-minimum $HOME/.config/BraveSoftware
/usr/local/bin/brave-browser
rm -rf $HOME/.config/BraveSoftware
Now absolutely everything except your preferences and bookmarks are deleted every time you close the browser.