I feel that the third sentence, “ He had many principles. The first one was let NO politician tell him what to do” has a more determined feeling too it, that, whatever "He" will do, one thing he will not let occur is to let ANY, NOT EVEN ONE politician tell him what to do. the third sentence can be corrected further to be written as follows: “ He had many principles. The first one was **to** let NO politician tell him what to do” ^ the sentence above is most accurate and most formal from my point of view. or “ He had many principles. The first one was, "Let NO politician tell [him] what to do.” the third sentence is written with the most vigorous feel to it.

The first two sound the most natural. The first one is more grammatically correct because in the second one you split the infinitive (inserted the adverb "not" between the infinitive "to let"). However, doing so is very common, and you will often see it in writing and hear it in conversation.

2nd one allows you to more emphasise "NOT" a bit more but 1st and 2nd are correct

Either the 1st or 2nd one sounds okay in English. The 1st is more technically correct.

2nd one slightly better