Bear in mind that only around 10% if lucky of the total fruit will stay on the tree to maturity by October and some may drop around June.

You more then likely purchased the tree in a 3 to 5 gallon pot from the nursery and now have transplated it in a 20 gallon container or urn as mentioned. Not knowing what type of soil medium you used to ensure excellent drainage and if you have fertilized the tree with either a liquid soluble or pelletized product as of yet would help in possibly determining the cause of the blackened fruit. The leaves look good to me with no yellowing so that is a plus in your favor.

What variety is the Satsuma? [Miho, Seto, Owari]

I put in pelletized fertilizer when I planted it and again about a week ago. I'm not sure what variety the Satsuma is and I'm afraid the soil may not be the best. I didn't replace the soil that has been in that urn since before we moved into this house, except for what went in the hole I dug for the tree. Does any of that info help with further diagnosis?
Thank you so much for your help.
Agi

I'd suggest that the planting medium is part of the problem. When aiming for success with a container grown plant, one needs to spend a lot of attention to the growing mix.

The soil DOES need to be the 'best', so to speak. Or your plants will retaliate, believe me. ;-)

so, at this point, how do I fix that problem? Can I just swap out soil without stressing out the tree?

It might stress the tree, but that is preferable to it staying in a poor soil. Changing now could save the life of your tree in the longrun. Mixes with some peat, perlite and bark work well for me, with about about 30% of it being peat, and the rest the free draining stuff. Do some research on soil mixes - believe me, its better to change it now than wait until the tree starts showing stress from the soil.