Kansas City doesn't hand you a nonstop to England's second city, and honestly, that's fine once you understand how to route it well. There is no direct service between MCI and BHX, so business class from Kansas City to Birmingham requires a connection, typically through Atlanta or Detroit on Delta, Chicago O'Hare on American, or Chicago/Newark on United, before crossing the Atlantic to a UK gateway like London Heathrow with a short onward hop or rail/car link to Birmingham. Delta's transatlantic metal, especially on the A330-900 or A350 out of Atlanta or Detroit, features their Delta One suites with sliding doors, which in my experience is the most consistently comfortable way to bridge the pond on this itinerary. American routes you through Chicago and typically partners with British Airways for the transatlantic leg, putting you in Club Suite with direct aisle access if you're lucky enough to snag that metal rather than an American-operated 777 or 787 business cabin.
The real insider move here is padding your Chicago or Atlanta connection to at least two hours, not because of tight gates, but because these hubs' international terminals are where you want to burn Delta Sky Club or Admirals Club access before a long transatlantic sit — arrive early, eat well, shower if the lounge offers it, because BHX itself has minimal business-class lounge infrastructure worth waiting for on arrival. I've found that flying into Heathrow and taking a connecting flight or the Chiltern Railways/Avanti rail link into Birmingham city center often beats forcing a same-day regional connection, especially after an overnight flight when you're running on interrupted sleep and want control over your own pace.
Seasonally, January through March and again in October and November are when fares on this corridor soften considerably, as Midwest-to-UK demand dips outside summer holidays and December's crush. Given how fragmented this routing is across three carriers and multiple connection points, your best strategy is setting up fare alerts and letting automated tracking do the heavy lifting so you can pounce the moment premium cabin space opens at a rate worth booking.