We were told at the time of Bishop Faure's consecration in 2015 that he was one of the original THREE candidates chosen by Archbishop Lefebvre.

The then Fr Faure declined and suggested Fr de Galarreta in his stead.

The other two were Frs Williamson and Tissier de Mallerais.

A delegation of Swiss faithful approached the Archbishop and asked for a Swiss bishop, given that they had played so important a role in the establishment of the Society. That is when Fr Fellay joined the party.

So I guess the original four submitted to Rome were: Frs Richard Williamson, Tissier de Mallerais, Jean-Michel Faure, and another unknown.

It has been said emphatically that when only one bishop was being permitted by Rome, the initial choice of the Archbishop was Fr Richard Williamson. Perhaps that was in the biography written by Dr White. I've never heard anything about Fr Black being in the mix, certainly not as 'the English speaking bishop', that would contrary to what is generally held that Bishop Williamson was always the Archbishop's first choice.

Why did the Four ask that their excommunication be lifted if they thought the pope wasn't even
holding the faith but a new religion altogether?

I am interested in the part that "A delegation of Swiss faithful approached the Archbishop and asked for a Swiss bishop, given that they had played so important a role in the establishment of the Society. That is when Fr Fellay joined the party".

Out of all of the Swiss priests why Fr Fellay? Considering he was only consecrated in 1982 and was seen part of the modern wing of the SSPX even before the 90s internal battle of the SSPX. (Bp Fellay tried to oust Bp Williamson out of Winona for District Superior of the Philippines) 

I also agree that the initial choice of the Archbishop was Fr Richard Williamson, no shadow of a doubt without that, he was the hand picked successor after all.

However, the Fr Black assertion has been a thing I have heard whispered in SSPX Chapels across the UK and a reputable old guard priest hinted to such. I think it is somewhat credible owing to his closeness to +ABL and his longevity as District Superior for the SSPX in the UK and then Australia. 

Why did the Four ask that their excommunication be lifted if they thought the pope wasn't even
holding the faith but a new religion altogether?

Unofficial.
  Google translate
Declaration of the General Chapter 2006
Excerpt. Let me know if you want the whole doc. .
The old SSPX demanded that the DECREE be removed, which really means that the entire SSPX considered it null and void. Fast forward to Jan. 21, 2009, and I think the Sspx thanked Pope B...? For what??

The position of Sspx has always been Recognize and Resist.( R & R).
********
http://www.virgo-maria.org/D-FSSPX/Chapitre-General-2006/declaration-du-chapitre-2006.html#_ftn1

"...In its discussions with Rome in recent years, the Society has been able to ascertain the validity and necessity of the two prerequisites [1] it has requested, which would bring great benefit to the Church by restoring at least some of its rights to its own Tradition. Not only would the treasure of graces enjoyed by the Society be brought out of the closet, but it would also provide the remedy so desperately needed for healing to the Mystical Body.

If, after their accomplishment, the Society awaits the possibility of doctrinal discussions, it is still with the aim of making the voice of traditional doctrine resonate more strongly within the Church. Indeed, the contacts it maintains occasionally with the Roman authorities have the sole purpose of helping them reclaim the Tradition that the Church cannot deny without losing its identity, and not of seeking personal advantage or of reaching an impossible, purely practical "agreement."..."


[1] La liberté entière et sans conditions pour la Messe tridentine et le retrait du décret d’excommunication des quatre évêques de la Fraternité.
(Complete and unconditional freedom for the Tridentine Mass and the withdrawal of the decree of excommunication of the four bishops of the Fraternity. Td.)

https://www.catholicnewsagency.com/news/14881/pope-benedict-lifts-excommunication-of-bishops-ordained-by-lefebvre
Vatican statement:
"...The Holy Father has been inspired in this decision by the hope of reaching the soonest possible full reconciliation and complete communion,” the statement concludes.

https://laportelatine.org/formation/crise-eglise/sacres-1988/histoire-des-sacres-episcopaux-1988-annonce-des-sacres

+ Lefebvre was asking for ONE bishop from Ratzinger, taken from the ranks of the SSPX. May 5, 1988 seems that was still the basic plan.

(then THREE)
"...Moreover, the Fr. of Chalard having confirmed the intention of the founder of Ecône de consecrate three bishops on 30 June, the cardinal asked him to transmit an invitation to return to Rome. A new meeting is then scheduled for 24 May..."

( Original French text."...De plus, l’abbé du Chalard ayant confirmé l’intention du fondateur d’Ecône de sacrer trois évêques au 30 juin, le cardinal lui demande de transmettre une invitation à revenir à Rome. Une nouvelle
rencontre est alors prévue pour le 24 mai...")


...since "we thought that you would give us the means to continue and develop the works of tradition, especially by giving me some coadjutors, at least three, and also giving the Roman organism a majority to Tradition". For it is always a question of remaining “without any progressive and conciliar influence”. Archbishop Lefebvre did not change this point.

(again THREE bishops)...
Finally, Archbishop Lefebvre takes matters into his own hands:

“That is why, to our great regret, we are obliged to ask you that, before the date of ¹ June, you clearly indicate to us what the intentions of the Holy See are on these two points: consecration of the three bishops applied for 30 June and majority of the members of the Tradition in the Roman Commission. Without replying to these requests, I will publish the names of the candidates that I will be devoting on 30 June with the assistance of S. Exc. Monsignor de Castro Mayer. My health, the apostolic necessities for the growth of our works, no longer allow for additional time.”

“That is why we will give ourselves the means to continue the work that Providence has entrusted to us, assured by the letter of His Eminence Cardinal Ratzinger of 30 May, that the episcopal consecration is not contrary to the will of the Holy See, since it is granted by 15 August. We will continue to pray that modern Rome, infested with modernism, will return to Catholic Rome..."

Finally, the Press conference June 15 ?, 1988, + L speaks of FOUR bishops.
We must place the events that are taking place today that are going to happen in the near future – particularly the episcopal consecration of four young bishops on 30 June – in the context of our difficulties with Rome, not only since 1970, since the founding of Fraternity, but since the Council...

(Il faut placer les événements qui se passent aujourd’hui et qui vont se passer demain – particulièrement la consécration épiscopale de quatre jeunes évêques le 30 juin – dans le contexte de nos difficultés avec Rome, non seulement depuis 1970, depuis la fondation d’Écône, mais depuis le Concile.)...

I don't think +Fellay became liberalized because he was paying the bills.  I think he either chose (or was placed) in such a role because it's an "admin role" where he would have a quicker way to become Superior General.  In a small association such as the sspx, what is the quickest way to reach the top leadership?  Some role where you can meet many people and become everyone's friend.  Being the bookkeeper allowed +Fellay to be a politician and (eventually) squirm his way to getting the votes necessary for leadership.

The other 3 bishops were busy working, not politiking, or having meetings.  It would be interesting to know who supported +Fellay staying in Econe and paying the bills.  This seems like something a seminarian should be doing.  Not a bishop.  If you find out who supported the move, you'll find out the people who eventually supported him being the leader.  You'll also find the same people who are pushing the "deal with new-rome" and all the changes since +ABL died.  +Fellay didn't get where he did, alone.

http://entremont-autrement.blogspot.com/2012/04/le-retour-decone.html

"...Bernard Fellay's Coat of Arms published during his savage ordination on 30 June 1988, which had earned him the lifting of excommunication on 21 January 2009 clearly show his motto: Spes nostra,..."


Les armes de Bernard Fellay publiées lors de son ordination sauvage le 30 juin 1988 qui lui avait valu une excommunication levée le 21 janvier 2009 font clairement apparaître sa devise: Spes nostra,



Bishop B. Fellay's Coat of Arms


......................

Below: Variant published by French District.

I notice that the motto "SPES NOSTRA" , is lacking. You have to love that French humor! Maybe I'm reading too much into that, but then why two?


Pray for Bishop Fellay.