You need a visa for PPL but not for ATP.
Since you have an ICAO ATP the easiest would be to take the FAA written exam and take the FAA checkride.
Gets you an independent FAA certificate.
(3) Holds either a foreign airline transport pilot license with instrument privileges, or a foreign commercial pilot license with an instrument rating, that—
(i) Was issued by a contracting State to the Convention on International Civil Aviation; and
(ii) Contains no geographical limitations.
(e) Meet the aeronautical experience requirements of this subpart that apply to the aircraft category and class rating sought before applying for the practical test;
(f) Pass a knowledge test on the aeronautical knowledge areas of §61.155(c) of this part that apply to the aircraft category and class rating sought;
(g) Pass the practical test on the areas of operation listed in §61.157(e) of this part that apply to the aircraft category and class rating sought; and
61.153 is what you are looking for;
Electronic Code of Federal Regulations: