I don't intend to be mean, but I had to laugh when I read what you have gone through to buy a ticket for a routine itinerary.
I laughed because I have had similar, but not as bad, experiences with Brazil's airline websites.
Maybe the Brazilians found a way to communicate with the beyond and then contracted with the Keystone Cops to program their airline websites.
When you use TAM on the U.S. site, it often does not work correctly. If you try GOL, it often rejects U.S. credit cards as payment and also insists on a CPF number, even from non Brazilians. You might want to add .br to the web address. For example, instead of www.tam.com you might get better luck if you enter www.tam.com.br. The same is true with GOL. However, even if you add the br, you often run into problems.
The poorly programmed Brazilian airline websites explain why so many Brazilians voluntarily stand in line for hours to buy a simple airline ticket.
If you notice when walking in large cities, there are countless travel agencies and most of them are busy. In the U.S. a travel agency is almost as rare as a hen's tooth. Most people in the U.S. and in other countries use websites to buy airline tickets. In Brazil, only sometimes.
TAM does have an office in Miami where you can call to book a ticket. However, the prices when purchased through the U.S. TAM offices are more than double what you will pay IF, and only IF you can make the reservation on line or in person at one of the many travel agencies in Brazil.
Boa sorte -