Isle of Wight
Yes, the copy is still an 'instant source' so Pyromancer's Swath will increase the damage.
706.2. When copying an object, the copy acquires the copiable values of the original object’s characteristics and, for an object on the stack, choices made when casting or activating it (mode, targets, the value of X, whether a kicker cost was paid, how it will affect multiple targets, and so on). The “copiable values” are the values derived from the text printed on the object (that text being name, mana cost, card type, subtype, supertype, expansion symbol, rules text, power, toughness, and/or loyalty)