Hi poolecw,
From what I've read,Black Mulberries are slow growers.
I have a Black Beauty that was planted about a year ago and it's not too much taller now.It has lots of leaves and some fruit,but has grown maybe 6-8 inches.
Maybe it wlll grow according to the saying,"first year sleep,second year creep,third year leap".
The first year I used Citrustone,but now it's some 20-20-20 that's on hand,1st time Feb-March,2nd time May,3rd time late July-Aug.I don't think they are too particular about fertilizer and 10-10-10 is probably okay. Brady

I fertilized my mulberry with Ironite Plus to get them moving this first year. They all arrived as young whips and seemed to me to be stunted but since giving them the ironite each of the five have grown several feet and put out multiple branches. The ironite helped my trees put out the grow I was looking for and may work for you as well.

Well I stopped in at my local nursery today. The owner suggested that I use some basic tree and shrub feed. We'll see how well the young plant likes it.

What about water? Is one gallon per day too much?

Irrigation depends on how much evaporation there is and things like mulch.
If it's in the ground and mulched,there probably won't be needed as much as when in a container.
I usually feel down about three inches and if it feels dry,then water. Brady

I can testify that Morus nigra is slow-growing. Since 2010 mine have put on at the most 6 inches of new growth per season. This is a fraction of what Morus alba will grow in my climate, but the nigra gradually fills in much more nicely, producing a thick trunk and branches, and of course those way tastier fruits. I've never fertilized mine, only mulched, and the foliage has always appeared healthy.