The Solution

Again the gods came to Bellerophon's aid, suggesting that a lump of lead affixed to the end of the spear would have a decidedly deadly effect. Firstly, when thrust into the monster's maw, it would cause the Chimaera to gag. And secondly, when melted by the beast's fiery breath, it would trickle down into its innards and cause a fatal case of heartburn.