I also wanted to add that because of the current way you are pulling him around his barrel he is already trying to cheat you, like you said here "He does go around the first part of the barrel very well, sometimes he gets way to tight and then he doesn't want to finish through the second half of the barrel". This is the first signs of him already trying to cheat you. Do not allow him to pivot/roll back on his back feet on the back of the barrel or drop into the turn, keep his body rounded and his shoulder picked up the entire way around the barrel. It will also be beneficial if you make him complete his turns while he is still learning.
Here is a video that kind of demonstrates what I am trying to say. (do not judge to harsh this is a little old and I am not perfect my hand placement is high) but do you see how my mare finishes her turn almost completely before I ask her to more over and run out? This will help prevent your guy from cheat the end of his turns in the future and trying to run out before it is time or swinging out wide. (by the way I am the one in the blue shirt 2nd and 3rd goes)
(also this was the first show for both of these horse, the first duo was not the horse or the riders first show just first show together....this was not my first show but it was my mares first show)