No, it is not possible. All modern Edges stores activities (tracks) in the FIT file format. BaseCamp can read those files, but it can't delete/modify them.
You can delete them manually, but there's another problem: the Finder cannot access your Edge 1050 because it uses MTP...
Use OpenMTP to access your device. Note: only one program can access an MTP device at the same time. So close BaseCamp (and Express) before using OpenMTP.
