I think that Bruce is the mask in the sense that since his parents were killed, his life as Bruce Wayne was over. "The mission" was his sole motivator: avenging his parents by erasing crime. Everything he does is to advance the success of The Mission, and "Bruce Wayne", the person, has become a secondary or tertiary concern. "Bruce Wayne" is a convenient façade that can be used as a resource for The Mission. But in that sense, Batman is also a mask, a means to an end. Bruce, Batman, and everything surrounding them are tools to further The Mission.
That's why legacy has been so important to they Batman mythos in recent years: Prodigal, Beyond, Batman & Robin, Snyder's body clones, etc. It's all about securing and ensuring the success of The Mission.
Alternatively, you can see him being Batman and not Bruce Wayne (and not the other way around) because Batman is a product of The Mission, while Bruce Wayne is a product of... well, his parents. He cannot live a life as Bruce Wayne because the "real" Bruce died in that alley. He is Batman in the sense that is a vessel for The Mission.