Heat olive oil in a Dutch oven or a large pot over medium-high heat; cook and stir chorizo in hot oil until heated through, about 2 minutes. Remove chorizo using a slotted spoon to a bowl. Add chicken; cook and stir until chicken is browned and crumbly, 5 to 6 minutes. Season with chili powder and cumin.
Stir chorizo, onion, garlic, red bell pepper, black beans, and white beans into chicken; cook and stir until onion begin to soften, 5 to 6 minutes. Pour beer into chili; bring to a simmer. Add tomatoes, chicken stock, and tomato paste; simmer until heated through, about 5 minutes. Season with salt. Reduce heat to low and simmer until flavors blend, about 30 minutes.