It should be no surprise that Saudi Arabia is the country paying the most for U.S. military personnel to work for them:

Foreign governments have long advanced their interests in Washington by paying Americans as lobbyists, lawyers, political consultants, think tank analysts and public relations advisers. But the hiring of retired U.S. military personnel for their expertise and political clout has accelerated over the past decade as oil-rich gulf monarchies have splurged on defense spending and strengthened their security partnerships with the Pentagon.
Congress permits retired troops as well as reservists to work for foreign governments if they first obtain approval from their branch of the armed forces and the State Department. But the U.S. government has fought to keep the hirings secret. For years, it withheld virtually all information about the practice, including which countries employ the most retired U.S. service members and how much money is at stake.
To shed light on the matter, The Post sued the Army, the Air Force, the Navy, the Marine Corps and the State Department in federal court under the Freedom of Information Act (FOIA). After a two-year legal battle, The Post obtained more than 4,000 pages of documents, including case files for about 450 retired soldiers, sailors, airmen and Marines.
Washington Post
You can read more at the link, but the article did not mention any retired military officials working with South Korea or Japan. It appears to all be governments from the Middle East, Turkey, and Russia. The fact that Korea and Japan has USFK and USFJ probably means they don’t need to hire outside experts like these other countries are doing.