Too late for this year, but spring cabbage will work if you are not planting out your beans before 20th May.
Ditto overwintering chard, which usually goes to seed just as you wish to plant out beans.
For when your beans are growing, I grew a very satisfactory beetroot crop on the morning sun side of the beanpoles: transplanted outfrom modules in late May, then harvested October-early December.
I tried dwarf beans the afternoon sun side, but they did not germinate well.