OK. I've managed to get rid of the annoying update badge (using TimeMachine only) on my iMac Late 2013 running Catalina 10.15.7.
First, make sure you have a TimeMachine backup which includes GarageBand 10.3.5 (or make one).
Second, delete the GarageBand App** by dragging the App icon in Launchpad to the trash. Restart your machine and the red badge will no longer be showing in the App Store icon.
Third, now open a Finder window, select Applications and then enter TimeMachine.
Fourth, Select the latest backup and the finder window will show that last backup's Applications window.
Lastly, scroll down and right click on the GarageBand App icon and choose restore. when it asks "where to" select your Applications folder.
GarageBand will then re-installed and be ready to use.
Obviously, make sure you have backed up all your projects before attempting this.
** do NOT delete this App without a backup as you will NOT be able to re-download it from the App Store.