Hey Rob,
Welcome to the forum...
Gorgeous Lychee Tree!! It seems to be very happy.
I have two lychees in containers, a Sweetheart and a Mauritius. I didn't get any fruit this year, but went ahead and pruned them. At the same time I also repotted both up to 10 gal containers, pruning their roots, esp. the circling roots. They are both doing just fine and have flushed out nicely with all the warm weather we are having lately.
I have noticed that they don't mind being root bound as much. However, you have to keep up with the watering (sometimes daily in the heat of summer). They do like a lot of water, and if you let them dry up, you will loose fresh tips and leaves, and if not watered, you can loose the lychee plant within days. Just something to be aware of for container growing, esp. in our hot South.