I guess I would describe the noise as a knock. Upon release of the string. At first I thought it may be an overspined arrow slapping against the bow but after trying multiple spines the noise didn't go away.
Pull to full draw then let down slowly and listen for the "knock." If no sound, try a very heavy arrow and see if you get the sound. It probably seems like a small point, but does the sound happen "upon release of the string," or when the arrow leaves the string. It matters. Also, with bow unbraced, put the handle in your vice (pad with leather as needed) and push the limb forward and back and see if you detect any slop.
I'm thinking it's not fore and aft slop making the noise. Even a hinged bow with no latch is stable as long as the string is intact.
Above all, find out what's making the noise before you start throwing "cures" at it or give up and epoxy it.