The statement doesn't need to say it if the bank will issue you a credit advice stating that the transfers were international. Since they're able to tell you exactly how they were transferred, they should be able to issue this to you and you can use it for your extension.
The only one that can "certify" the birth is Thai family court. If you were not married when the child was born, you have to go through the court to get recognized as father. It's not a fast or cheap process. You'll need a lawyer. It's your only option.
Also, your congressperson cannot help with getting a foreigner a tourist visa to Thailand.
The only thing a congressperson can help with is submitting an inquiry to USCIS if a visa application is taking longer than the stated timeline to process, and even then it rarely helps. That's for immigrant visas like K1 fiance or marriage that take months or years to process.
You can do it up to 3 days before arrival. You cannot choose a date that's greater than 3 days away on the form. But you can choose a date that's 3 days, 2 days, 1 day, or today.