I like your idea of switching the chicken run and garden every year. If you switch the plots in the fall, then the chicken poo should have more than enough time to cool down for a spring planting.
Exactly what I'm thinking. In addition to giving the chicken poo time to break down, making the swap at the end of the growing season gives the chickens a chance eat the garden plants, including any weed or plant seeds you'd want cleaned out before your next planting.
The setup would also help break any disease/fungus cycles that can happen in a static garden.