So far, the WoodNet guys agree with me. Mulberry would not make a long-lasting ax handle. Here are some comments so far:
(1) No on the Mulberry. You can mark it with a fingernail.
Commercially made handles are almost all hickory, for a reason.
(2) The advantage of hickory over red mulberry is its modulus of rupture (breaking strength) is nearly twice as high, as it's elastic modulus (flexular strength). White ash sits in between and is often used for tool handles. Its rot resistance is poor, however, so it's not a great choice for anything that will be left in the weather.
(3) I have to agree with the weakness, softness, and ease of breaking. Mulberry is just not tough enough for an ax handle or a splitting maul. Hedge (osage orange) might work, but not mulberry. Even ash would be a better alternative or perhaps hard maple (not silver maple).