I'm far from a guru but with Brave open, go to the three little dots on the bottom right corner of your phone. Choose bookmarks, then choose the three little dots on the top right corner of your phone. That allows you to either import or export your bookmarks, hope that helps.

I just tried it and the bookmarks file exported to the downloads folder in the Files app. This is all with the newest version of Android.

Brave must have finally added that, I hadn't checked in awhile!

Thank you for making me try again!

In the meantime, I found this:

https://www.youtube.com/@EricMurphyxyz

I'm using Brave on a desktop - may be variations.

Go to settings - either via menu or little cog wheel bottom right.
Chose Privacy and Security on the left
Scroll down about 5 sections to Clear Browsing Data

and then check your third party cookies settings.

Just for some sites.

You may need to look for a cookie manager add-on so you can whitelist some sites with cookies and data you want to keep.

But it does not seem to be working for me correctly at the moment.

Arrgh.