I don’t know if these will help answer your question, but they might:
New PC - How to Copy Brave Custodian
How I keep my BATs after formating my computer?
My understanding (and I am probably wrong!), is if you have a verified wallet, you can transfer your BAT to that wallet and they will be saved. You would just need to reinstall Brave, create a wallet, and then link with the verified wallet that is holding your BAT. However, this will use up another lifetime slot.
If you are already at the max lifetime slots (4), then you need to submit an unlinking request (the link is in the first post but here it is again): Mattches Browser Support Attention
Hopefully, someone with more experience will respond and help. Good luck!