AUSTIN—Since 1975, Skeeter Miller’s The County Line barbecue restaurants have been dishing out ribs, brisket, turkey and sausage “so good you’ll want to get it all over ya,” their slogan says. Loyalists who have been coming since the beginning remember all-you-can-eat plates for $6.75.

Amid the pandemic and the fast-rising cost of meat, Mr. Miller has had to raise prices, including four times in just the past year, he said. When he pushed the price of the Big Daddy Platter—a hefty rack of slow-smoked beef ribs—to $32.99 from $18.99, he handed all the waiters sheets with his cellphone number to give to any customers who complained.