You can trap iguanas without having to hire trappers/exterminators. But hiring them would be a lot easier.
You can find more information on the Net. Plus one of our members lives in S. Florida and knows how to trap iguanas. He might respond to this thread.
But even if you remove all iguanas in your yard, others will move in to replace them. It is a tough problem, especially since you have children.
I have to advise you to wear tough long jeans whenever you go out and might encounter that big one. Iguanas are known to escalate their interactions with humans. he may only be wanting to claim you as his love object, but males are often rough with their mates, and female iguanas have much tougher skin than you do.