If you want to change clothes at some point during the trip, that's going to have to factor into the equation as well. Some airlines are real sticklers about bowling equipment and weights. And they can refuse heavy things or force you to pay a surcharge. That four-ball carrier seems like a good idea until they say no. Friends from your area can tell you what airlines are better than others.
Some names pop up over and over again, but things change over time, and from airport to airport.
So ask around. If it's just a fun tournament, or you're not quite Team USA calibur, think of bringing just a few pieces of equipment. Something that hooks, something that doesn't. If it's an important tournament, and you need a large arsenal, considering shipping a few via UPS ahead of time, and have the hotel store them for you. Your pro shop can help you with boxes.