Honestly, you'll have the least amount of trouble if you simply replace the carb with a good used one from another quad. What model of Bayou are you working on? A 220? A 300? Either way, these machines were made virtually unchanged for years, so finding a good replacement won't be hard.
Sure, you could install an aftermarket Mikuni or Keihin carb, but do you really want to go through the hassles of figuring out the proper jetting and needle spec that works best on your engine? I'd rather not, especially when that 4-stroke engine won't benefit much (if any) performance-wise from a bigger or different carburetor installation.