I am extremely prejudiced toward the King James Bible of 1611. I have found it to be superior in every case I have studied in detail and have been blessed from God with a simple, yet strong, faith in its very words. Therefore, I have no problem in saying that the King James is correct and the New Scofield revision is wrong. However, that being said, let me add a little background.
The text of the New Scofield Bible is not Scofield's position. C. I. Scofield died in 1921, long before a committee made the revisions found in the 1967 revision called the New Scofield. Since Scofield did sometimes correct the text of the King James Bible, I cannot say what he thought about the adding of the phrase in 1 Corinthians 9:20. But it was not in his text and he had no note to that effect either.
You point our that the New Scofield notes state concerning this phrase: "The KJV omits this italicized clause, which appears in all important Greek mss." According to the critical Greek New Testament of the United Bible Societies (which agrees with the addition), this phrase is not found in quite a number of Greek Manuscripts. In fact, according to their notes, the King James reading is "the reading of the majority of the Byzantine witnesses." This reading is also found in the 5th century Syrian Version, the 6th century Ethiopian Version, and the 5th century Georgian Version. It is quite a subjective call to say that the omitted phrase is found "in all important Greek mss." At least I admit that I am prejudiced. These comments are made by scholars who claim an objective and disinterested approach.
The phrase itself may be true. In a real way, Paul was no longer under the law. To the Romans, Paul stated, "for ye are not under the law, but under grace;" and, "shall we sin, because we are not under the law, but under grace?" (Romans 6:14-15). In the second statement, he clearly included himself with those who were not under the law. However, the statement is an addition in 1 Corinthians 9:20 and should be omitted as it is in the King James Bible. At the very least, it is superfluous and is not needed. If he is determined to appear "as under the law" to the Jews, it would only make sense that he is not under the law now.