Let fish stand at room temperature for 20 minutes. Divide mint-chive butter into 2 bowls. Season cavity of fish with salt and pepper, and brush generously with half the mint-chive butter. Tie together with kitchen twine to close. Brush with oil, and season with salt and pepper.
Heat a charcoal grill to medium. Lightly brush grates with oil. Sear lemons, cut sides down, until caramelized, about 3 minutes; transfer to a serving platter. Brush grates to clean.
Grill fish, covered, about 12 minutes. (Check on fish once or twice to make sure heat is even and fish is not charring.) Open lid, and flip fish carefully using 2 heavy spatulas. Cover, and grill until skin is crisp and fish is flaky when pierced with the tip of a knife in the thickest area, 8 to 10 minutes. Transfer fish to a platter using spatulas. Brush fish with remaining mint-chive butter. Let stand for 10 minutes; untie. Garnish with mint.