Bran Stark took the Iron Throne. GoJo acquired Waystar Royco. But who will rule the Texas Renaissance Festival? That’s the battle at the center of “Ren Faire,” HBO’s new docuseries set to air in the network’s coveted Sunday-evening slot during the two weeks ahead of “House of the Dragon” Season 2.

“It sounds like that’s what their calculus was, to put it as close to that show as possible,” series director Lance Oppenheim says, laughing.

“Ren Faire” may have the knights and knaves of “Thrones,” but the power dynamics running through it more closely resemble “Succession.” Produced by Josh and Benny Safdie’s Elara Pictures, the three-episode doc centers on George Coulam, the iconoclastic octogenarian founder of the “nation’s largest Renaissance theme park,” which welcomes about half a million guests annually to carouse among costumed entertainers while munching turkey legs and cheering on jousters.

The festival...