Great news — this feature was just implemented with the release of v1.75.175. You can use Ctrl+Shift+S on Windows and Linux and Cmd+Shift+S on macOS to take a screenshot in Brave.
Going to go ahead and close this thread shortly as well as the feature is now present in the browser. Thank you for everyone who submitted a request.
EDIT: Note that you can also change the keyboard shortcut for taking a screenshot in brave://settings/system/shortcuts.