Bristol's regional airport has no meaningful widebody long-haul service from the American West Coast, which is precisely why savvy travelers booking business class from Los Angeles to Bristol route their itineraries through London Heathrow, connecting hubs on the East Coast, or occasionally Amsterdam, rather than expecting a nonstop. American, Delta, and United each offer viable one-stop routings: American via its Philadelphia or JFK hub into Heathrow with a domestic-style hop onward to Bristol on a partner, Delta via its Atlanta or JFK gateway with Air France-KLM or Virgin Atlantic connections, and United via its Newark hub, which has historically been the strongest single-connection option to the UK. The realistic answer to "how do I fly business from LAX to BRS" is: fly a US carrier's flagship transatlantic business product to a European hub, then take a short regional connection into Bristol.
I've flown this general corridor enough times to know the real prize is the transatlantic leg itself. United's Polaris seats out of LAX or its connecting hubs offer a proper lie-flat suite with direct aisle access on 777 and 787 aircraft, while Delta's One Suites—now standard on many of its long-haul widebodies—deliver sliding privacy doors that make a red-eye out of LAX genuinely restful. American's Flagship Suite, especially on its 777-300ER, remains one of the more spacious hard products among the three, though availability into UK-bound connections can be tighter in summer. My insider tip: book the transatlantic segment as a separate positioning flight if award space is scarce on a single itinerary—repositioning through JFK or Newark a day early often unlocks Polaris or One Suite saver space that doesn't exist on a same-day connection, and it gives you a buffer against Bristol's notoriously fog-prone regional airport in winter.
Seasonality matters more on this route than most transatlantic pairings, since Bristol's connecting flights thin out considerably outside peak UK holiday travel. January, February, March, October, and November consistently offer the best combination of fare value and business class award availability, as demand dips after the fall foliage crowd and before spring breaks begin. Given how much these fares swing week to week depending on hub routing and partner space, setting up automated fare alerts is the smartest way to catch a premium seat before it disappears.