The DLP Endpoint Agent currently does not support the Brave browser by default. The Brave executable can be added to Application Monitoring, which will add detection for HTTP. However, there is still a lack of HTTPS detection.
Up to DLP Version 15.7, there is currently no default detection for the Brave browser in the DLP Endpoint Agent.
An Enhancement Request has been submitted to Product Management to add support for a future release. The request can be followed up on through a Sales or Account Manager.