Msgr. Lefebvre and his bishops did not appeal the sentence against them by the Congregation for Bishops. In any case, John Paul II confirmed the judgment in "Ecclesia Dei".
Bl. Pius IX's words about a former schism seem pertinent:
Since this does not please the neo-schismatics, they follow the example of heretics of more recent times. They argue that the sentence of schism and excommunication pronounced against them by the Archbishop of Tyana, the Apostolic Delegate in Constantinople, was unjust, and consequently void of strength and influence. They have claimed also that they are unable to accept the sentence because the faithful might desert to the heretics if deprived of their ministration. These novel arguments were wholly unknown and unheard of by the ancient Fathers of the Church. For "the whole Church throughout the world knows that the See of the blessed Apostle Peter has the right of loosing again what any pontiffs have bound, since this See possesses the right of judging the whole Church, and no one may judge its judgment." [. . .] But the neo-schismatics have gone further, since "every schism fabricates a heresy for itself to justify its withdrawal from the Church." Indeed they have even accused this Apostolic See as well, as if We had exceeded the limits of Our power in commanding that certain points of discipline were to be observed in the Patriarchate of Armenia. (Quartus Supra)
It's all right there: appeals to the state of necessity, claims of unjust excommunication (apparently without a trial), claims that the Pope has exceeded his powers, and so on. Bl. Pius IX gave short shrift to such arguments. Pius VI, of blessed memory, has the same teaching: "the right of ordaining bishops [. . .] cannot be assumed by any bishop or metropolitan without obliging Us to declare schismatic both those who ordain and those who are ordained" (Charitas). He later prohibits the appeal to the "pretext of necessity".
Heck, New Rome didn't even give him a trial.
47. Likewise, the proposition which teaches that it is necessary, according to the natural and divine laws, for either excommunication or suspension, that a personal examination should precede, and that, therefore, sentences called "ipso facto" have no other force than that of a serious threat without any actual effect,-false, rash, pernicious, injurious to the power of the Church, erroneous. (Pius VI, Auctorem Fidei, DZ 1547)
What nonsense. What need was there to appeal? The fix was in. Do you think the Archbishop was stupid? The Holy See was abandoning the traditional faith--as this article clearly shows. It was time to act--and he did.