Washington DC might be a possibility, but I'd only invoke it if your journey between Central America and Asia will be taking you to the USA for at least a week or so, not flying in the US domestically during that time, and you can send from inside the USA. I'd tend to use an agent, rush fees, and express courier services to expedite things, even at additional cost. If not going through and staying in the US for awhile and/or if you want to ship from South or Central America, then I'd recommend sending it back to an agent in your home country by intl FedEx or DHL and have them rush process and send back while you stay put in a specific area. It's cheaper to hole up for a while in S. America than it is the USA.
It sounds like if you are not able to get satisfaction from one of the honorary consuls in South America or Mexico, then your only other options are the above strategy, or to route through South Korea or Japan (or Hong Kong) for a few days and get a visa in one of those locations in-person.
Whichever way you look at it, you're going to be spending money either for the passport to travel somewhere, or for changes in whatever airplane routing you already have, to do it in person.