The diamond in lathe bits is actually man made (diamond dust compressed) and requires a diamond impregnated wheel. Check with this guy. He sharpened one for me that was chipped badly and it turned out to be the best bit I ever had.
GW diamond cutting co.
4100 w. Kennedy Blvd.
Tampa, Fl.
813-287-0068
It takes a while to get it done, So I bought another bit and Got mine sharpened. When the other one came back, I used the new one as a backup. Never needed it though. If you are going to buy a new one, the Carter bit is the best.