This is why you should make photocopies of all official documents (various passport pages, visas, birth certificates, etc). Carry one set with you, separately from the passport itself. Leave a set with a trusted friend/relative at home who can fax them to you when needed.

You can also take digital photos and mail them to yourself at an email address you can access from anywhere.

BTW the airline can't do anything. You have to go to the US embassy, assuming you lose a US passport. You may or may not need a local police report; I would try the embassy first. Copies of visa etc are in case you are questioned by local authorities. It will still be a major inconvenience, but having all this info will make the process smoother.