Red Mulberry makes a fine bow. The only wood that I know that doesn't discriminate between the heartwood and sapwood, either way or a mixture of the two offers great results. The density of both sapwood and heartwood are very close on this wood. The bend properties I can't vouch for other than both seem about equal to me. I agree with the comments on sapwood integrity through the drying process, be careful how you dry it.
I just posted a mulberry bow not long ago, there are some tips and comments on drying that you may want to consider (I think, if that's the one). Actually you can look back through my posts and see examples of a few bows built of heartwood and sapwood for poor examples.
I call Mulberry my little sports car wood. Its low density and high elastic properties will perform with the best of them.
One of my favorite woods by far.