Will blackcurrant bushes (ribes nigrum or ribes ussuriense) fruit in USDA zone 10b (border with zone 9) in an urban container garden with SW sun?

As Greg Tatar noted for most black currant varieties that zone is just to warm. It really needs a temperate climate.

The blackcurrant (Ribes nigrum), also known as black currant or cassis, is a woody shrub in the family Grossulariaceae grown for its berries. It is native to temperate parts of central and northern Europe and northern Asia, where it prefers damp fertile soils and is widely cultivated both commercially and domestically.

• Many cultivars are hardy to -40F (-40C). These cold-natured plants seldom produce well in warmer climates where temperature often top 90F (32C).
• Currants grow best in summer humid, cool regions with great winter chilling. The bushes stop growing and leaves tend to scorch when daytime temperatures reach 87 degrees F.

Furthermore some varieties are self-fertile. Those that are not would require a second variety to ensure good fruit set. They are not conducive to being restricted to a container either. They can grow 3-6 feet wide and tall before they bear fruit.

Sorry to sound so negative but this is just not going to work.

Blackcurrant - Wikipedia

Black Currant Grow Guide

Growing currants and gooseberries in the home garden

How to Grow and Use Currant Bushes