I would try taking a piece that's 3/8 x 7/8 and round over 4 edges, then drill a hole slightly smaller than the tang and drive it in. A groove the same size will loosen fairly quickly. If the tang is tapered then I would try to drill to different depths in steps to help prevent splitting. Putting a ferrule on the end helps too but you would need to turn that section round to fit. Another thing that would help prevent splitting would be to take some sturdy braided fishing line and wrapping the end which could be carved down to near round. (Start by making sawcuts and removing most of the waste that way. It only takes a few minutes to do the job that way.) "Whipping" the line is the best method and there is/are likely some vids on youtube on how to whip the end of something. Once you're done add a couple coats of some varnish or glue to toughen the wrap.
Someone I consider a master woodworker once told me that a master woodworker is not someone who never makes mistakes. He is someone who is able to cover them up so that no one can tell.