Hi,
Climbing beans are probably not the best ones for that. Use the short growing ones and tie those to the tree, or they will compete for sun.
I plant all around my figtrees, be it tomatoes, lettuce, cucumber, cabbage.
The theory is that you water the vegetables, and all excess water gets caught by the (fig)tree. The same for the fertilizer.
The problem comes when the canopy of the tree gets too strong. Then, pumpkins are what is left to plant, as they will benefit from running from sunny places to more shady places.
Just be careful as those climbing veggies can bend and even crack branches on the trees, especially in windy and rainy weathers .