Combine cabbage, radishes, cumin, cilantro, and lime juice in a large bowl. Add salt to taste and transfer to a serving bowl.
Sauté lamb, garlic, and oregano in a large frying pan over high heat, breaking up meat with a wooden spoon until very little pink remains, about 5 minutes. Drain excess fat. Add chile powders, beer, vinegar, and salt to taste and cook, stirring, until most liquid has evaporated, 2 to 3 minutes more. Transfer to a serving bowl.
Set out tortillas and toppings in separate bowls and let everyone make their own tacos.
Make ahead: Cook meat and prep toppings the night before.