Your main considerations for purchasing your trailer probably fall into two areas: flat bed versus enclosed trailer; and single-axle versus tandem axle.
As a bbq competitor, the enclosed trailer is definitely the way to go based on my personal experience. It offers you a dry place to sleep, a warm place to prep meat and turn-in boxes, and you can lock it up and feel secure when traveling, or if it's parked in a remote location for an extended period of time.
That leaves you with a decision about whether to purchase single-axle or tandem-axle trailer. I've put together a chart (see below) based on my personal observations and thought processes that might help you decide.
I feel that the tandem-axle trailer offers much more stability on the highway, a certain amount of safety if you should have a flat tire (four tires are better than two), and I think they are constructed better overall versus most single-axle trailers I've seen.
To illustrate my point, I was following a single-axle trailer today. I won't mention the manufacturer, but as we approached a railroad tracks at 20 mph I noticed that the driver towing the trailer made no attempt to slow down at all. He didn't touch the brakes even slightly. The trailer actually bounded completely off the ground and then each of the tires took turns skipping off the ground until both tires were safely on the ground after we'd traveled 100 ft or so past the railroad tracks. I hope the trailer was empty because if there was anything in it, I don't even want to guess what shape it's in at this point. I actually thought the trailer was going to turn over. In that situation, a tandem-axle trailer would have been much more stable and all four tires would not have lost contact with the ground.
If you're still undecided, talk to some folks that have driven both types of trailers out on the highway. I think you'll find out real quickly that most people that tow trailers for distances farther than just around town or back and forth to Lowe's and Home Depot are going to recommend the tandem axle trailer every time.
The heavier trailer does require a bigger tow vehicle, but the safety and security are worth it.