The KJV is translated from the Textus Receptus from the Greek texts of the Computensian Polyglot, from Erasmus, from Beza 1598 and Stephanus 1550.
Professor Frederick Scrivener, in 1881, produced a learned text in which, from his extensive knowledge of the manuscripts involved, he compiled a Greek text (which sits beside me as I write) being the actual Greek text underlying the KJV.
This Scrivener text also shows the additions, the omissions and the alterations made by Westcott and Hort in their new Greek text of 1881 from which the Revised Version was translated. (It was not translated from the Textus Receptus.) This is the basis of the Nestle Aland text which is translated into English as the NASB version.
In I Corinthians 9:20, Wescott and Hort have added the words :
μη ων αυτος υπο νομον
These words are not in the Textus Receptus and therefore not in the KJV.
They are also not In Robert Young's Literal Bible 1862 nor in Green's Literal Bible of 1993.
These are the words which your other versions translate.
Paul expressly says in Romans 7 :9 :
I was alive without the law, once, but when the commandment came, sin revived and I died
and the commandment, which was ordained to life, I found to be unto death.
Paul clearly states his experience of coming under legal conviction. He is clearly subject to the effects of law. It slaughters him when it says :
Thou shalt not covet.
For, he confesses :
I had not known lust except the law had said 'Thou shalt not covet' [Romans 7:7 KJV]
This opens up a greater issue : that of ancient texts (specifically Vaticanus and Sinaiticus) and whether these two (which disagree between themselves in 3,000 places in the Gospels alone - see Dean John Burgon in his book 'Revision Revised' 1881) should be given undue preponderance simply because of antiquity.
(Just for comparison, the several versions of the Textus Receptus differ from one another in about 289 places throughout the 27 books of the New Testament and none of these differences is of doctrinal significance.)
Dean John Burgon's argument is that ancient manuscripts did not survive as they were often read and therefore became worn (they were made of skin) and had to be re-copied.
Dean John Burgon argues that the ancient manuscripts in the Vatican and in the monastery of Sinai were corrupted texts and although useful as background evidence should not be viewed as having vastly more weight than all the other evidence.
Each person has to examine these issues for themselves and has to be persuaded in their own mind about what version of the Bible they read.
It could be argued that the words should be included and that Paul is saying that he is no longer under law but under grace.
Which then gives many people a problem who say that believers are still under law despite being believers.
So, an interesting question indeed.
Personally, I don't mind whether it is included in the canon of scripture or not, if there is a difficulty with absolute certainty. Paul was brought up an Hebrew, strictly ; he came under legal conviction as he makes clear ; and he was also delivered from the law as is abundantly demonstrated in Romans and Galatians.