"the text of Lumen Gentium MUST be a coherent extension and illumination of Tradition."
Why? Have you read it?
While there are many excellent passages in it, these are often hedged about by ambiguities, circiterisms and novel propositions which have no basis in either Scripture or Tradition. You don't have to read further than chapter 1, para 1 to find them.
I know that it is a coherent extension and illumination of tradition for the same reason that I know that the documents resulting from the Council of Trent and all ecumenical councils are coherent exstensions and illuminations of tradition. It's that simple. Some might call it simplistic, but I call it a radical faith in the power of the Holy Spirit acting in and through the Church. That being said, I respect the Catholics who disagree with my position as long as they exhibit a genuine piety and love for the Church.
Ch.1. Paragraph 1
1. Christ is the Light of nations. Because this is so, this Sacred Synod gathered together in the Holy Spirit eagerly desires, by proclaiming the Gospel to every creature,(1) to bring the light of Christ to all men, a light brightly visible on the countenance of the Church. Since the Church is in Christ like a sacrament or as a sign and instrument both of a very closely knit union with God and of the unity of the whole human race, it desires now to unfold more fully to the faithful of the Church and to the whole world its own inner nature and universal mission. This it intends to do following faithfully the teaching of previous councils. The present- day conditions of the world add greater urgency to this work of the Church so that all men, joined more closely today by various social, technical and cultural ties, might also attain fuller unity in Christ.
What is it about this paragraph that should give concern to the faithful Catholic?