It seems that on this particular tree, there is somewhat of a naming controversy. (Imagine that...) It's always been marketed as Morus nigra but several places have taken to just calling it Morus Sp. I have seen some people who claim to be experts saying it is not nigra. Anyhow, you may be on to something with the naming...
And apparently there are several species/varieties being sold as "dwarf everbearing"
(now parroting internet research gleanings, including Wikipedia)
Added to that many people call a mulberry tree that has black mulberries a "black mulberry tree". While this may be a good description it is not necessarily a good species identification, many Morus species can have black or very dark fruit (including white mulberry) but it gets perpetuated by people who don't know better and call it M. nigra.
To add to the confusion several species hybridize easily, and several plants of other genera are called mulberries.
AND
M. rubra and M. nigra are not real easy to differentiate. One source said the only positive way to identify was by observing the nuclei with a microscope, M. nigra has a high polyploidy and the large number of chromosomes is obvious. The same author claims there is no dwarf or everbearing variety of M. nigra. (
mulberry varieties)
Pheww!
I think I might be able tell whether or not the one at the store is M. alba. If it is alba it might be worth messing with.