What doctrine do they reject? You make claims "the Vatican" does not even make. "The Vatican" doesn't even call Protestants Protestant, they're separated brethren, don't you know. Who died and made you pope? Calumny is a sin you know.
No one died and made anyone here pope BUT conclaves, through the work of the Holy Ghost, made Karol Wojtlywa into Pope John Paul II and Josef Cardinal Ratzinger into Pope Benedict XVI. Catholics obeyed JP II and obey Benedict XVI accordingly. Non-Catholics do not. Non-Catholic Christians are either Orthodox or Protestant. No one that I know of accuses SSPX of being part of Eastern Orthodoxy (that would indeed be calmny against our Orthodox brethren in Christ). That leaves Protestant as the category applicable to SSPX (with appropriate apologies to our separated brethren of good intent and our prayers for them in having to share the category with SSPX) and the excommunicated leaders of SSPX bear a very strong stylistic resemblance to Luther, at al., in their attacks on the institution of the Church and on the papacy. Protestants have a moral advantage over SSPX in that they do not claim dishonestly to be Catholic while refusing Catholic authority. If the shoe fits (and it certainly does), you must wear it. You are not entitled to invent your own facts.