As a routine, I backup bookmarks - actually export the Brave Browser (iOS) Bookmarks (and tabs, as bookmarks).
The exported Bookmarks are a bookmarks.html file, that I place outside the Brave Browser (iOS) installation path - ie NOT in the Brave sub-folder of the Files app.
Next, I completely remove the Brave Browser (iOS) installation, from the iPhone.
Next, a complete shutdown of the iPhone:
Exit / Quit everything ← meaning, completely, the Apple iOS way - a complete “hard”/“force” restart of the iPhone (and same, if the work applies to an iPad):
https://support.apple.com/guide/iphone/force-restart-iphone-iph8903c3ee6/ios
https://support.apple.com/guide/ipad/force-restart-ipad-ipad9955c007
Then run a fresh installation of Brave Browser (iOS).
And import the bookmarks.