M.nigra consistently gets the nod for best flavor. I've never had the opportunity to sample the fruit - and don't know how well, if at all, it would adapt to container culture.
I suppose my first choice for a 'tender' variety to attempt pot culture would be 'Pakistan' - I've grown it in-ground here in KY, but it winter-killed back to the ground every year, so no fruit. It's offered by a number of different nurseries - I know Edible Landscaping & perhaps Raintree or OneGreenWorld have had it in the past; probably others available as well.
Just Fruits & Exotics may also have some other 'tender' varieties that would be potential contenders.

Here is a link that might be useful: JF&E Mulberries

You might also want to check out Burnt Ridge Nursery, they have Illinios Everbearing as well as a couple other varieties.

I grew up eating "wild" mulberries and now I enjoy IE mulberries as well as a white variety in a friend's yard. (good for drying, they're really too sweet to enjoy too may of them fresh). I also have never had the opportunity to try M. Nigra (though I know where there is a tree near my parent's in Fl, though it never seem to be bearing when we visit).

~Chills

I tasted the Pakistan type when I was visting a friend in Malibu. The fruit are huge and very sweet but they weren't a whole lot better than I E (I might be predjudiced, however). I like your idea of trying it as a container plant- let us know how it goes.

I also have seen them in the Edible Landscaping catalogue. The quality of there mulberries have been a bit spotty in orders I've made, however. I still think they are a pretty good nursery.

The best thing about Pakistani mulberries is that they are extremely easy to pick. I still like Persians the best for taste. Just be aware that Pakistanis are very frost sensitive. They push new growth very early. I have lost the entire crop several times because of late frosts. I don't know about growing them in containers are they are very fast growers, at least for me. Even in a 20 gallon pot you will need to do a lot of root pruning.

I'm not as knowledgeable about tropical mulberries as vilandry. I'm not even sure if I was eating Pakistanis or Persians.

Check out Burnt Ridge Nursery, Ediblelandscaping.com, and Rollingrivernursery.com for mulberries.