Usually with bunnies, they recommend you cut it, then let it age for a week or more before giving it to your pet, and even put it in a low oven for an hour or so, just in case of any parasites or bugs it might have. Are you certain no pesticides have been used on the tree?
Its on safe bird wood lists, the usual rule of thumb is if its safe for birds its safe for chins but the two members in your thread are smart cookies, so based on that and their concerns I may pass, even thought the wood could be safe.