I've been using Brave for over a year. As I understand it, it's all encrypted before it leaves the browser, so they couldn't track it even if they wanted to. The FAQ says:

Brave Payments uses a technology called Anonize that permits anonymous-but-accountable transactions. Here's an analogy from the real world: someone walks into a polling place, identifies themselves, and gets a ballot and an envelope. They go into a voting booth, mark the ballot, put the ballot in the envelope, and seal it. They then drop the envelope in a ballot box. The people running the polling place and counting the ballots know that each person putting an envelope in the ballot box is authorized to vote, but they aren't able to determine which envelope in the ballot box goes with which person. In the case of anonize, a special branch of cryptography called "Zero Knowledge Proofs" are used to get the same functionality

I assume that won't change when they switch from Bitcoin to BAT.