Point one: quit worrying about your 4yo. As long as this is a domestic flight, a ticketed child does NOT need ID for TSA. They will ask her what her name is when you approach security, but that's it.
Point two: your DH should NEVER be putting legal ID in checked baggage. That's a really good way of asking to become an identity theft victim.
Point three: if you do not have ID, TSA will still let you fly, but you'll have to go through a secondary security search. This includes a patdown and a careful scrutiny of everything in your bags, both carryon and checked. If it does happen, it will save you grief if you don't check or carry on any bags in your own name -- let them go as belonging to other members of the party who have ID.
The above only applies to domestic travel. If you are overseas and lose your passport, you will not be allowed to re-enter the US without one. The thing to do in that situation is to go immediately to the US consulate wherever you are visiting, and have someone from home fax a copy of your passport photo page to the consulate so that they can replace your passport. I have a copy of my passport page kept in my HR file at work; someone is always there to be able to send it to me on a weekday. I only need to resort to that if I also lose the extra photocopy of it that I keep in my money belt, or the USB drive that I keep on my keychain, which also contains a PDF scan of the photo page.