I'm planning to purchase some mulberry trees, then cut them very short in order to get some fruiting shrubs. Does this sound like a bad idea to make them easily picked?
There are about 10 - 16 species in the genus Morus and there is significant variation in ultimate height even within species. Morus nigra is the most species most commonly cultivated for its fruit and is usually grown as a tree. It is generally recommended not to prune them (they can bleed a lot) and they tend to fruit better without intervention.
I suspect that the tree would take coppicing quite well but I suspect that the resulting growth maybe shy of fruiting for some years after the cut and of course would need to be performed at regularly to stop them achieving their full height. You may find that you are forever removing the best fruiting material. That said they can be trained as an espalier where appropriate pruning leads to the formation of fruiting spurs. So a restrictive pruning regime is compatible with fruiting, but it would not be as simple as simply coppicing them.
If you need a small fruiting shrub, there are a number of dwarf cultivars and hybrids that may be more suitable for your purpose such as Mulberry ‘Charlotte Russe’ which forms a suckering shrub to around 1.5m tall, as does _Morus rotundiloba 'Mojo Berry'. To my mind, these are not a patch on a mature Morus nigra in fruit. From a fruiting point of view, the joy of mulberries is the sheer exuberance with which mature trees fruit. It is quite an incredible bounty. Given that they are beautiful as trees, why not let them grow as nature intended? Sometimes our interventions don't help matters. If you need something smaller, there are countless similar fruits (Rubus - the bramble genus) that are perfectly suited for growing underneath trees. Ribes (the currants) also like a bit of a shade.
A traditional "mulberry bush" is 4 to 5 feet high which is easy enough to pick.
You need to be aware they probably won't fruit at all till they are 10 years old, and the fruit season is only two or three weeks. But old trees and bushes look very nice, if you plan to keep them for 50 or 100 years.
That is possibly why in Britain they were historically associated with the royal family - nobody else could afford the time, space, and labour to grow such a "luxury" crop!
They were also food for silkworms. There's a nice story about king James I who planted four acres of mulberry trees in the grounds of Buckingham Palace, to start a British silk industry. Unfortunately, he planted the wrong species of mulberry and the silkworms couldn't live on them. However some of the original trees are still there, 400 years later.
Aside from fruit and silkworms, they can be a health hazard for people with asthma, hay fever, etc, because the male trees produce huge amounts of pollen. They are banned in some US cities for that reason.
I don't know if they will fruit if cut back hard. In one of the gardens I look after there was a large mulberry tree that fruited prolifically, year after year. Anyway, long story short, about twelve years ago this tree was felled (I forget the reason - it may have split). But it regrew from the stump. Very slowly. It's now about as tall as I am. But it hasn't fruited yet. I planted a replacement mulberry a few yards away just after the original was felled, and this new one even now barely fruits. If you like mulberries it might be best just to plant a tree. And wait.
They won't do much in the form of a "bush". Mulberry trees are of the genus "Morus" (see Wikipedia link). Depending on the age and type of mulberry trees that you have they will tend to grow into a tree and are usually quite attractive - as a tree. However, there are smaller ornamental cultivars as well, and yours may be one of those. There are both male and female trees. The female trees don't take too long to start producing fruit - maybe three or four years.
The fruiting season is around late June through July and the fruit don't all ripen at once. The berries ripen a few at a time, but the fruit is soft and over ripens quickly. The are quite good to eat, but you have to harvest them as they ripen and eat/process them immediately. The fruit also attract lots of song bird that relish them as well. Many of the unpicked ripe berries will fall to the ground so some people consider them to be "messy", so you should choose your tree location wisely. With all of the bird droppings and falling berries, the underlying soil might be expected to become fertile and rich which might enhance the undergrowth as you suggest. The Wikipedia link below has more information.