Sorry, but no cigar. You just identified Lebanon with Hezbollah. Try reading what you bolded again.

Nothing is 'mere'. — Richard P. Feynman

You want Bush, or any American leader, to work with Hezbollah (funded by Iran)?

In Iraq, the US works together with Da'wa and SCIRI.

I just don't see how that would work

I suppose you're right: it isn't working in Iraq either.

Is there a non-militant Hezbollah wing?

I don't know about "non-militant", but "non-military" there is. Like Hamas or the Muslim Brotherhood, the reason these organizations are successful is that they provide the social services that our friendly dictators don't provide.

I just haven't heard of one

I suppose you haven't been listening to the right sources. Try wikipedia for a start.

The civilian wing of Hezbollah runs hospitals, news services, and educational facilities and participates in the Lebanese Parliament. Its Reconstruction Campaign (Jihad al-Bina) is responsible for numerous economic and infrastructural development projects in Shia-populated areas of Lebanon.

Nothing is 'mere'. — Richard P. Feynman

As Miguel pointed out, you're confusing Lebanon with Hezbollah.  Hezbollah may be centered in South Lebanon, but the Lebanese are not all Hezbollah members.  And, yes, most of these groups have non-militant wings.  They participate in politics.  What frightens me is that we have a new government in Lebanon that, prior to this crisis, seemed to be showing some promise.  Israel had an anti-Syria neighbour to its North, but it may well drive the population (and the government), out of fear, to support groups like Hezbollah.  Rather than help Lebanon clean up Hezbollah, Israel is destabilising the Hariri government.  And bear in mind that the destruction of terrorism is not a foreign concept to Hariri, whose father was murdered with a car bomb suspected of being planted by the Syrian govenment -- this, according to Wikipedia.

Be nice to America. Or we'll bring democracy to your country.