Black mulberry trees are small with dense growth. All varieties are pretty similar from what I gather. I have Black Beauty, which has delicious fruit produced over a long season (6 or more weeks in later summer). The birds have a hard time finding the berries, which is good, but so do you, which is not so good. Because of the 'everbearing' habit of black mulberry, you never get many berries at one time it seems.

In terms of shape, this depends a great deal on where it is grafted. Grafted low it can form a shrub, but they are typically grafted at about 5 feet above the ground, so they make a small, full, rounded tree with a 5-foot trunk. Sounds like you might want to search for a low-grafted one and prune it to a short tree form.

For productivity you might consider a named variety of white mulberry, but they tend to be larger more vigorous trees than the black.