As you can see from people, there’s the “official answer” and the “unofficial.” Technically the only backup & restore for BAT is to have our devices connected to Uphold or Gemini. This said, you can transfer it over on devices such as Windows.
Users often suggest just one or two particular files but Product Manager (Rewards) with Brave suggests you don’t just copy partial files as you have a risk for problems. Instead, move your entire Brave folder.
I link to it all and give some better explanation in my FAQ at -Retired- PSA: User FAQ - Support & Answers
Make sure to check the portion entitled I'm getting a new device, how do I backup my BAT/information? OR I need to reinstall, how do I save my BAT/information?