I haven't used the Burley, but have used Adams, AlleyCat, and a few other cheaper trail a bikes. One thing that I have noticed about many of these, is that younger children are unable to reach the peddles to actually peddle. This is not only frustrating for the child, but caused quite a bit of instability and discomfort.
The Wee-hoo that I sampled at the store did not have this problem. It was easily able to go short enough for small legs. I was able to set it up for my 3 year old to peddle nicely. It's full back seat also allows small children to fall asleep during a ride. For longer rides the sleeping option is very nice. It has storage pockets that make it easy to carry snacks, toys, or other needed supplies. Carrying supplies has proved to be a challenge with some of the trail a bike products, although with the Burley I assume you could use panniers.
I have heard (although I cannot confirm this) that Burley is no longer making these and that has made replacement parts difficult to come by and is also inflating the price. In the time frame that I have been biking with kids (roughly 5 years) Burley has stopped producing these at least one other time. During that time period the Piccolo and Kazoo sold for high amounts, much like they are selling now. However, when the company starting producing them again, the prices came down dramatically and many people had a bike that they had spent more for, than could be purchased new. I think that I remember this product selling for around $250.
I really love the Wee-hoo product and if money wasn't an issue would have one for my son. Can you take your son down to the store to try both out and see what would work best for him?
We were able to get a compact Adam's trail a bike $150. After shortening the seat post, that is now working for our son.