Lord Drinian is the captain of the Dawn Treader, a loyal and able-bodied seaman who knows a great deal about maritime navigation. It's important for us to recognize that Caspian himself is not the captain. Although he is the King, and he makes decisions for everyone, he defers to Drinian's expertise when it comes to details of shipboard life. Sometimes Drinian shows his stubborn streak, such as in the episode of Goldwater Island, when Drinian wants to steer the landing boat in one direction and Caspian overrides him. The narrator tells us that Drinian "had had an anxious day with the weather yesterday, and he didn't like advice from landsmen. But he altered course" (8.32). So Drinian does think for himself and has normal human feelings, like irritation, but he doesn't let that get in the way of following orders.
We also see Drinian get irritated toward the end of the novel, when Reepicheep jumps overboard to fight the Sea People:
"Drat that mouse! [. . .] It's more trouble than all the rest of the ship's company put together. If there is any scrape to be got into, in it will get! It ought to be put in irons – keel-hauled – marooned – have its whiskers cut off." (15.27)
As the narrator explains, Drinian doesn't actually dislike Reepicheep; he's just worried about him and the rest of the crew. Throughout The Voyage of the Dawn Treader, Drinian's loyalty to King Caspian and the ship's crew remains unwavering.