Unless there's some miracle product like some epoxy that can fill the groove in the fret and restore normal use then I'm not sure. Without that then replacing the frets, is really it. If the guitar has a bolt on neck then perhaps buy a whole replacement neck, which might be cheaper in the end. If you haveafret with a groove in it, there's likely others that are past their time, you'll probably be up for replacing them all. Otherwise having a few newer frets mixed in with old will look odd.
*Anyone have an idea of the costs involved???
Pretty sure welding is out of the question, any welding or even soldering will impart a lot of heat into the metal and being so small it will fall apart, or bend beyond repair.
Filing it a little will have helped to remove some of the burr on the edges of the groove and perhaps help just a little in rounding the edges of the groove to make it not 'feel' so serious. But as a fix, it's not going to work because too much and you'll reduce the height of 12th fret down until the string will rattle against the 13th fret.