Whom might Travis be referring to as his royal majesty?

The line that you are looking for can be found about halfway through the story. Eckels is listing off all of the massive animals that he has supposedly hunted and killed at previous times in his life. He admits that he is shaking knowing that he is about to come face to face with a Tyrannosaurus rex. Travis then says "ah" in order to get everybody's attention. The group stops talking and everything goes quiet enough for Travis to whisper his next line of dialogue to the entire group.

Travis raised his hand. "Ahead," he whispered. "In the mist. There he is. There's His Royal Majesty now."

Travis isn't talking about a person or a human king of any sorts. Travis is referring to the T. rex. In present times, the lion is the King of the Beasts; however, millions of years ago, the T. rex could have been given this same title. The text then gives a description of the beast that leaves no doubt that the animal is a mighty predator. It must have been a sight to behold because Eckels announces that it is an impossible beast to kill.

"It can't be killed," Eckels pronounced this verdict quietly, as if there could be no argument. He had weighed the evidence and this was his considered opinion. The rifle in his hands seemed a cap gun. "We were fools to come. This is impossible."
https://www.worldstory.net/en/lion.html

Travis, their safari guide as they travel to the dinosaur age, refers to the Tyrannosaurus rex as his royal majesty. This is an allusion to the idea of the T-rex as the king of the dinosaurs, the mightiest of all the great lizards.
The Tyrannosaurus rex will soon come into view. Bradbury describes it as thirty feet taller than all the trees and calls it an "evil god." Bradbury's description of the T-rex emphasizes how mighty and awesome this creature is, far bigger than any animal the men would see in the present day. It is a colossus that weighs ten tons, and Eckels becomes convinced it is too big to kill. This fear of the T-rex leads Eckels to panic and leave the metal Path, which in turn changes history.