Thanks for that information, that changed my thinking. You are essentially looking for a cover crop during the dry season that the chickens can forage on. Whether the chickens eat it and poop it out or you turn it under when planting, that is a really good way to improve the land. The more "compost" you can work into the soil the better. That's not just for the nutrients but it improves the tilth (workability) of the soil whether you have a sandy or clayey soil. One downside to your rainy season is that certain nutrients, especially nitrogen, are water soluble. They leach out of the soil, leaving it low in those nutrients. You may need to find a way to supplement the nitrogen especially. I don't know what is available to you but with all those animals compost or compost tea may be an organic option.
Chickens grazing like that should clear off the vegetation or at least most of it. But they are unlikely to totally take out the roots. You are probably 4 to 5 degrees from the equator, it is going to be hot and humid in summer but will never get cold enough to freeze in the winter. Your ideal cover crop would be an annual, not something that will come back from the roots. I don't know what those are in your climate. Clover is usually considered an outstanding cover crop and forage crop but some varieties of clover may be an annual in certain climates but a perennial in others.
I don't know what crops will work best in your climate. Often there is no ideal crop or mix. I'm trying to point out the traits I'd be looking for.
Good luck!