OK I think I found a solution and it had nothing to do with Brave exactly.
Turns out a system cleaning app had deleted a necessary file at:
~/Library/Application Support/Google/Chrome/NativeMessagingHosts/com.apple.passwordmanager.json
Even though I’m now exclusively using Brave, that file in that folder is apparently necessary for the extension to run, even with Chrome uninstalled.
For anyone else with this issue, you need to make the file manually with this as contents:
{ "name": "com.apple.passwordmanager", "description": "PasswordManagerBrowserExtensionHelper", "path": "/System/Cryptexes/App/System/Library/CoreServices/PasswordManagerBrowserExtensionHelper.app/Contents/MacOS/PasswordManagerBrowserExtensionHelper", "type": "stdio", "allowed_origins": ["chrome-extension://pejdijmoenmkgeppbflobdenhhabjlaj/"] }