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Arvin's patch leaves a lot of non bending wood below the patch to keep the handle section stable.
I would be surprised if your patch doesn't turn loose, you removed a LOT of the strength of the handle, I hope I am wrong. Even non bending handles stress and bend some even if it is a small amount. I have tried patches in the limbs, the patches would hold up for a couple of years and then turn lose.
I glued patches into this this handle just of see if I could get away with it because the rest of the stave was really good osage, I bamboo backed the bow to add a another layer of stability to the handle. I sold the bow cheaply, 15 years later it surfaced in an archery shop in Chattanooga, still in shootable condition
I started with this;