No dice, it still doesn't work. I tried with all toggles active in Filesystem section but none seem to make it work.

I tried adding the location for the Brave Flatpak in VS Code Flatpak with Flatseal.

It now shows the device but I get an error when launching the app for debug from VS Code.

Launching lib/main.dart on Chrome in debug mode...
[CHROME]:/var/lib/flatpak/app/com.brave.Browser/x86_64/stable/55eab7c3b790510e9175f8a064f16c84972163599c064dc833eff81bec751652/export/bin/com.brave.Browser: line 2: /usr/bin/flatpak: No such file or directory
Failed to launch browser after 3 tries. Command used to launch it: /var/lib/flatpak/exports/bin/com.brave.Browser --user-data-dir=/tmp/flutter_tools.LCRXFZ/flutter_tools_chrome_device.IXGKQA --remote-debugging-port=44639 --disable-background-timer-throttling --disable-extensions --disable-popup-blocking --bwsi --no-first-run --no-default-browser-check --disable-default-apps --disable-translate http://localhost:33631
Failed to launch browser. Make sure you are using an up-to-date Chrome or Edge. Otherwise, consider using -d web-server instead and filing an issue at https://github.com/flutter/flutter/issues.
Exited

I'm not sure what I need to modify to make VS Code launch Brave as a device in debug mode.

I'm doing my part