Mulberry is very good, you'll have to use the heart wood, like you would with osage. Mulberry has much thicker sap wood than osage, so you'll need a log of at least 10'' diameter, if you are planing to split it, anything less than 10'' and there just won't be enough heart wood for 2 decent bows.
Another option, if you find a log thiner than 10'' is not to split it down the middle, so you don't halve the heart wood, but just take a small wedge off of it, this should leave you with a workable size pice of heart wood.
Good luck.