How can an organization with a strong “made in America” identity compete in the global marketplace?

The number one rule in successful competition is this: Find your comparative advantage. What do you do better, faster, and/or cheaper than anyone else?

As is well known, wages are considerably higher in the US and other First World countries than they are in Third World countries such as India and Nicaragua. If you assume that a product has fixed quality and requires a fixed amount of worker-hours to produce, then (aside from shipping costs) it's obviously more efficient---and thus more profitable---to produce that product in a Third World country. Thus, a company that sells itself on "Made in the USA" would not be able to compete on that basis.

But these are not the only differences between these countries. Wages in the US aren't simply higher by some weird quirk or bad policy; they are higher because US workers have more skills and better education than workers in India. Technology is also better in the US; US workers may be able to use production methods such as automation, computer-aided design, or 3D printing that workers in India would not be able to do. Combining these factors, US workers might be able to produce many more goods per worker-hour, or goods of much higher quality, than workers in India. Thus, the goods could be cheaper because workers are so much more productive, or their higher quality could make people buy them even if they are more expensive.

Some goods also vary more in quality than others. Your shirt may say "Made in India", but I'm guessing your car doesn't---it was probably made in the US, Germany, or Japan, where workers are skilled and high-tech factories are available.

Finally, there's always American nationalism. A somewhat more cynical approach would be for the US business to really capitalize on American nationalism to get people to buy products that say "Made in the USA" even if they cost more and aren't actually any better. There is actually a well-known "puzzle" in economics about this, called the "home bias in trade puzzle": People are much more likely to buy goods from people in their own country or similar countries than they are to buy goods from more different countries, even if the goods are the same quality and much cheaper. Personally I don't find it all that puzzling: Humans are nationalistic.