It appears to me that you are basing the criteria for "precision of translation" on details of the doctrine of salvation which you understand from the KJV. To put it more directly:
1. The KJV says "X" therefore I believe X'.
2. The NASB says "Y" which is different from "X".
3. Because X' is the truth, the KJV is superior to the NASB.
In other words, that is circular reasoning.
Umm...
“Precision in translation” based from what the context is (internal evidence), and the proper usage of the terminology.
Now for external evidence for the usage of “begotten”, I will be using the non- bias online etymology bringing us the very thought of the KJ translator. Here what the etymological perspective of the word “begotten”:
begotten (adj.)
"procreated," late 14c., past participle adjective from beget.
beget (v.)
Old English begietan "to get by effort, find, acquire, attain, seize" (class V strong verb, past tense begeat, past participle begeaton), from be- + get (v.).
Sense of "to procreate" is from c. 1200, generally used of the father only. Similar formation in Old Saxon bigitan, Old High German pigezzan, Gothic bigitan "to get, obtain." Related: Begot; begotten.
Online Etymology Dictionary
Thus the usage of the word ‘beget/begotten’ was from the 13 ce, long before the English KJV of 17 ce. which generally “Begotten” was already used of the ‘FATHER’ only.
What about “born/bear”
bear (v.)
Old English beran
"to carry, bring; bring forth, give birth to, produce; to endure without resistance; to support, hold up, sustain; to wear" (class IV strong verb; past tense bær, past participle boren), from Proto-Germanic *beran (source also of Old Saxon beran, Old Frisian bera, Old High German beran, German gebären, Old Norse bera,
Gothic bairan "to carry, bear, give birth to"), from PIE root *bher- (1) "carry a burden, bring," also "give birth" (though only English and German strongly retain this sense, and Russian has beremennaya "pregnant").
Old English past tense bær became Middle English bare; alternative bore began to appear c. 1400, but bare remained the literary form till after 1600. Past participle distinction of borne for "carried" and born for "given birth" is from late 18c.
Many senses are from notion of "move onward by pressure." From c. 1300 as "possess as an attribute or characteristic." Meaning "sustain without sinking" is from 1520s; to bear (something) in mind is from 1530s; meaning "tend, be directed (in a certain way)" is from c. 1600. To bear down "proceed forcefully toward" (especially in nautical use) is from 1716. To bear up is from 1650s as "be firm, have fortitude."
Online Etymology Dictionary
So a simple question and that may require simple observation: Did God the Father bear a child? Did He gave birth to our Lord Jesus Christ? Am so sorry, the Bible has nothing to do with that kind of theory/logical conclusion or derivation you wish to imply when the word used is “born again” in this given text.
Psalms 33:4
For the word of the LORD is right; and all his works are done in truth.