I also have these tungston carbide tool tips I need to turn handles for still. They came dull from the factory and need to be sharpened. Similar outer bevel.
Your only real options are the belt sander in the metal shop and the tool grinder wheels with the wood lathes. The former gives you a wide range of grit options, so you can control how fast you remove material. Whichever you use, I suspect you’ll need to find or make some sort of jig to hold the tool at the correct angles to get those bevels.