Add enough boiling water to cover, and set a small plate or bowl on chiles to keep submerged.
Let soak until softened, about 30 minutes.
Drain chiles, reserving 1 cup soaking liquid.
Preheat oven to 350F.
Place chiles, sugar, lime juice, and 1/4 cup reserved soaking liquid in a blender. Pure chile mixture, adding more soaking liquid as needed to form a smooth paste. Season pork shoulder generously with salt and spread paste over pork. DO AHEAD: Can be rubbed 1-2 days ahead. Cover and chill.
Let come to room temperature before continuing.
Heat oil in a large heavy pot over medium heat.
Add onion, garlic, bay leaves, oregano, coriander, cumin, and allspice. Cook, stirring often, until onion is soft, about 8 minutes.
Add beer; bring to a boil.
Add pork to pot; cover and transfer to oven.
Braise pork, basting occasionally with pan juices, until very tender, about 2 1/2 hours. DO AHEAD: Can be made 2 days ahead.
Let cool slightly. Chill uncovered until cold, then cover and keep chilled. Rewarm before continuing.
Transfer pork to a large platter.
Let cool slightly.
Pour pan juice mixture over pork; cover and keep warm. (Alternatively, shred the pork with 2 forks and pour pan juice mixture over pork.)
Meanwhile, working in batches, cook tortillas in a large heavy skillet over low heat until toasted, about 1 minute per side.
Transfer to a large sheet of foil; wrap to keep warm.
Serve pork with tortillas, radishes, both salsas, pickled onions, and chopped cilantro, encouraging guests to fill and garnish tacos as desired.