You can build up the nut slot with superglue and baking soda, lots of YouTube videos on that. Cutting an access hole in the pickguard sounds eminently sensible. However, as the bass came without an E string, I wonder if the neck is a little twisted at the headstock end? Not that there’s a lot you can do about that with a single truss rod neck, but it would explain the problem. Building up the nut if the truss rod is correctly adjusted would still be the way forward.