Make the achiote butter: Mash butter, achiote paste, garlic, orange zest, lime juice, salt, cumin, and chipotle into a smooth, brick-red spread.
Prep the snapper: Pat fish very dry, especially in the slits. Rub half the achiote butter over skin and cavity; reserve the rest for basting.
Preheat & oil grates: Grill high direct heat, ≈500°F.
Grill & baste: Lay snapper skin-side down, press flat with a spatula 30 sec. Grill 5–6 min until skin chars and releases. Flip carefully; baste flesh with remaining butter. Cook 5–6 min more, lid closed, until thickest part hits 135°F.
Rest & serve: Transfer to platter, spoon buttery juices over, and hit with fresh lime. Carve along backbone into big flaky slabs.