There is no contradiction between MA and UR or LG. The contradiction is rather between what the Church is really and truly teaching, and what you wish she would teach. It is reminiscent of Protestants who confuse the words of Scripture with their interpretation of Scripture.
Note the teaching on the necessity of full incorporation into the Church. Note how partial incorporation lacks the full benefit of the sacraments. Note how I do not pit one Church teaching against another as you do, but incorporate all teachings together into a doctrinal synthesis which neither denies the Fathers, Popes or Councils which comprise the constant and universal teaching of the Church. Note how you are attempting to make the Church teach something she has never taught before, and, in fact, has specifically taught against. And, finally, note how this will be my last entry in this thread.
http://www.rc.net/rcchurch/vatican2/index.html
http://www.rc.net/rcchurch/vatican2/lumen.gen
14. This holy Council first of all turns its attention to the Catholic faithful. Basing itself on scripture and tradition, it teaches that the Church, a pilgrim now on earth, is necessary for salvation: the one Christ is mediator and the way of salvation; he is present to us in his body which is the Church. He himself explicitly asserted the necessity of faith and baptism (cf. Mk. 16:16; Jn. 3:5), and thereby affirmed at the same time the necessity of the Church which men enter through baptism as through a door. Hence they could not be saved who, knowing that the Catholic Church was founded as necessary by God through Christ, would refuse either to enter it, or to remain in it.
Fully incorporated into the Church are those who, possessing the Spirit of Christ, accept all the means of salvation given to the Church together with her entire organization, and who--by the bonds constituted by the profession of faith, the sacraments, ecclesiastical government, and communion--are joined in the visible structure of the Church of Christ, who rules her through the Supreme Pontiff and the bishops.
Even though incorporated into the Church, one who does not however persevere in charity is not saved. He remains indeed in the bosom of the Church, but "in body" not "in heart."[12] All children of the Church should
nevertheless remember that their exalted condition results, not from their own merits, but from the grace of Christ. If they fail to respond in thought, word and deed to that grace, not only shall they not be saved, but they shall be the more severely judged.[13]
http://www.rc.net/rcchurch/vatican2/unitatis.red
In order to establish this His holy Church everywhere in the world till the end of time, Christ entrusted to the College of the Twelve the task of teaching, ruling and sanctifying.[10] Among their number He selected Peter, and after his confession of faith determined that on him He would build His Church. Also to Peter He promised the keys of the kingdom of heaven,[11] and after His profession of love, entrusted all His sheep to him to be confirmed in faith[12] and shepherded in perfect unity.[13] Christ Jesus Himself was forever to remain the chief cornerstone[14] and
shepherd of our souls.[15]
Jesus Christ, then, willed that the apostles and their successors the bishops with Peter's successor at their head should preach the Gospel faithfully, administer the sacraments, and rule the Church in love. It is thus, under the action of the Holy Spirit, that Christ wills His people to increase, and He perfects His people's fellowship in unity: in their confessing the one faith, celebrating divine worship in common, and keeping the fraternal harmony of the family of God.
The Church, then, is God's only flock; it is like a standard lifted high for the nations to see it:[16] for it serves all mankind through the Gospel of peace[17] as it makes its pilgrim way in hope toward the goal of the fatherland above.[18]
This is the sacred mystery of the unity of the Church, in Christ and through Christ, the Holy Spirit energizing its various functions. It is a mystery that finds its highest exemplar and source in the unity of the Persons of the Trinity: the Father and the Son in the Holy Spirit, one God. ...
Nevertheless, our separated brethren, whether considered as individuals or as Communities and Churches, are not blessed with that unity which Jesus Christ wished to bestow on all those who through Him were born again into one body, and with Him quickened to newness of life- that unity which the Holy Scriptures and the ancient Tradition of the Church proclaim. For it is only through Christ's Catholic Church, which is "the all-embracing means of salvation," that they can benefit fully from the means of salvation.
We believe that Our Lord entrusted all the blessings of the New Covenant to the apostolic college alone, of which Peter is the head, in order to establish the one Body of Christ on earth to which all should be fully incorporated who belong in any way to the people of God.
This people of God, though still in its members liable to sin, is ever growing in Christ during its pilgrimage on earth, and is guided by God's gentle wisdom, according to His hidden designs, until it shall happily arrive at the fullness of eternal glory in the heavenly Jerusalem.