Fly to Bishkek to visit friends, then fly to Mashhad. Take the train to Damascus. Taxi from Damascus to Beirut, stay for a couple of weeks, then fly to Istanbul and back to Tashkent. This all depends on Iran's willingness to give a family of Americans transit visas.
I'm thinking that train travel would be perfect for a two-year-old. I'd love to just take the train from Tashkent to Tehran through Turkmenistan, but it looks like that's currently impossible.
There's also the possibility of taking the train through Turkmenistan and crossing the Caspian Sea to Baku, which would be so cool, but the only way to get from Baku to the Middle East is by bus. And that doesn't sound like fun with a two-year-old.
It's hardly straightforward to get from Tashkent to Beirut.