Mmmm, cross posting.. (but think I answered your question in my reply to Parts below, EBM)
No, it's not mulberry but I cannot find the plant's proper name in my books either though. Well, not that it's that important now but I've been wondering about its name for longer and think the solution to find its name is next time I visit my plant supplier (= huge store) to look for it.
Don't think the branches will grow much thicker (diagonally) than 1 and 1/8 or 2/8 inch and there are too many 'disturbances' in the surface of the wood to make it good carving material. Might try some later.. not splitting it first then.
(um, bit too much blahiah for a little piece of shrub's wood maybe
but hey.. )(I just like wood/working with it)