You can get Rated Battleground information at:
/profile/wow/character/{realm}/{character}/pvp-bracket/rbg
If the RBG endpoint returns a 4xx status code, then the character has not completed any RBGs since the RBG endpoint was released. This endpoint dates back at least a couple of seasons.
The RBG endpoint is linked to from:
/profile/wow/character/{realm}/{character}/pvp-summary
If the PvP Summary endpoint does not link to the RBG endpoint in the brackets field, then the character has not completed any RBGs this season.
Both of these endpoints update on character logout.