No, you cannot
Captivating Melody (Complete Mage, p.40) is a general feat that says:
As a swift action before casting a spell, you can attempt a Perform check (DC 15 + the level of the spell you intend to cast). If you succeed, you can sacrifice one of your daily uses of bardic music to increase the save DC of the next enchantment or illusion spell you cast in the same round by 2. If the Perform check fails, you still lose one daily use of bardic music but gain no benefit.
As you noticed, bard's songs are either Supernatural or Spell-like abilities, but are not spells. On a similar subject that follows the same ruling, you may also not apply metamagic feats to supernatural abilities, as seen in this other answer. Nor abilities that improve or upgrade spells in general, as seen on this other answer.