As I know Basecamp cannot delete or edit anything on a nuvi, only on outdoor devices.
You need to copy the waypoints to your computer, edit them, then send them back to the Nuvi like you wrote.
You can delete favourites from Basecamp in nuvi, but when you use Basecamp to send favourites to the Nuvi, it does by creating a .gpx file and placing it in the GPX folder on the device. When the Nuvi starts up, it automatically imports any favourites from that file if they don't already exist.
So if you delete a favourite on Nuvi, then restart it, the old .gpx file that Basecamp sent previously is still there, and the old favourite is read from it. The solution is to delete the .gpx file (probably named Temp.gpx) from the device, and THEN delete the favourite.