Agree with Pearl. Walnut backing isn't a great choice.
If you made a walnut backing it would have to be PERFECT, preferably sawn from the outside of a log "under the bark" like a selfbow would have, and even then I dunno.......maybe on a very lightweight bow? Save the walnut to make nock overlays, recurve stiffening overlays on the belly, wedges, and handles.
But, take a look at your hickory boards again. Hickory is tough stuff, and if the grain is good you can make a flat sawn, quartersawn, or even rift sawn backing. Can you re-orient the board diagonally, and then again at a bias (with a jig, please, for safety) and get a backing out of it? Can you maybe get a perfect 38" piece and splice the backing in the handle?
I've seen bows backed with red oak or other "lesser" woods do fine, but the grain was really good, and walnut just seems to want to separate under tension, even in selfbows.