Since his rise to power in 1922 - and increasingly so with time - Mussolini had been the center of the Italian political life, for whomever wanted to be politically active under the fascist rule.
His decline in the years of the war, leading up to 1943, had been sharp, both politically and physically. Yet to many his fall had been a surprise. Behind the initial shock, though, the galaxy of competing powers and instances that used to gravitate around the leader somehow survived. The german decision to put the Duce back in - limited - power offered them both a dilemma and an opportunity.
Mussolini was again the leader but he had no longer the strenght or the will to personalize the italian politics. At the same time the Pope and the King were out of the picture. This offered the chance to push forward some of the issues that had been kept in the background - often for reasons of real politique, or in their view merely opportunistic.
But why? Only the most delusionals thought at the time that the war could be won yet - and even in that unlikely situation, it would not have been an italian victory. Wasn't that a good time to give up on fascism, acknowledge defeat or maybe get rid of the black shirt worn to appease the regime by mere conservatives?
Before going on, the question as stated leads to a clear no as an answer. There was almost no room for the pursue of any policy in the RSI - certainly far less than before. Any action needed indeed the german approval.
But there was a way of working around things on which I will go back later...
Nonetheless the correct issue - for the people involved - was not wheter they could but wheter they were supposed to act as if they could or rather give up.
I try to avoid the moral issue in general but - given the nature of the excerpts proposed - I feel compelled to notice that I do not believe persevering with fascism for sake of coherence to be a sound moral ground.
One of the leading figures of the newborn RSI was Alessandro Pavolini - the first secretary of the Republican Fascist Party - a fanatic, if this simplistic designation is to be forgiven; chosen by Mussolini for his energy and committment. He describes by himself the reasons behind his decision to join the PFR.
Every revolution can not be made if not with blood. And is righteous the blood poured to secure new rights; but unrighteous that poured while hiding... If the source of any new right is revolution ... than it would not deserve that name if it gave [the future generations] the doubt on its nature. It is true now that the future of the fascist revolution is tied to that of the war, whose unfavourable end it could hardly survive. But it would not even leave that minimal heritage that every movement leaves to the survivors, if it were not imprinting its strenght trough its will and committment, only possible through the energy of the actions undertaken and the risk for their authors to expose themselves to the vengeance of the reaction.
For Pavolini the social issue is not the central point. What matters more is the chance to restore, to resurrect the "fascist revolution" by committing to its most intransigent values. In this regard the government and the party must act as a Comité de Salut Public with the purpose of putting deserters, saboteurs to the wall.
Mussolini had picked Pavolini already in Rastenburg, putting him in officially in charge of the party with his radio broadcast of September 15th.
He was now tasked with building a new state, a new government, a new idea that could possibly appeal to the people and hide somehow the effective lack of power and autonomy from the german occupants.
The republican choice was obvious - if not for Mussolini's own belief, for the german pressures. So was the rethoric of the return to the fascism of the first years in a program that again made a central point of corporativism.
But those issues had not been actually implemented in almost 21 years of fascist rule and Mussolini did not actually believe they were going to be. His main concern was finding people for his government who needed to be at the same time: loyal to him, competent enough and not too obscure in order to gain some recognition among the population, palatable for the germans.
Mussolini - in a sudden unrealistic hope to again act as a peace mediator - tried to persuade the former ambassador in Moscow Augusto Rosso to take the Foreign Ministery but the latter refused. In the end the Duce was forced to take the position himself.
For the defence Marshal Rodolfo Graziani was picked, despite being misliked by everybody involved, after the refusal of Marshal Enrico Caviglia who was offered at first the command of Rome. His position on the current situation was stated in September 14th privately as follows:
I foresee that Hitler will put back Mussolini as chief of occupied Italy. If I were to stay in the government, I would be put aside without care. In any case I could do nothing for Italy. Even if the german authorities would leave me in charge, I would have to be their servant. My government would be undermined and powerless and I would be forced to resign.
Graziani had been himself uncertain wheter to accept or not the offer. In his later recollections he would say that the decisive factor was the clear threat by german ambassador Rahn that - if the italian leading figures were not to show their support to the regime - Italy would have been treated as Poland had been.
The belief that they were avoiding a worse fate for the population was certainly a contributing factor to the support given to the new regime. But the hoplesseness of the military situation and the inability of the RSI to lead any authonomous policy - besides police action against partisans - confined the new state to that of a pure puppet state.
This did not prevent Mussolini - at least during the first months - from attempting to build a wider consensus platform around his regime. He still believed in the possibility to obtain a moderate popular support, in order to salvage something out of the desperate situation; but despite his activism those initiatives were doomed.
On one side he tried to bring back to the party those who had left in disappointment after the instituzionalization and growing conservativism of the mature regime. He even attempted openings to old opposers in order to create a wider front in hopes of an armistice.
On the other he opened the doors to the new generation of fascists and here he was a bit more succesful. The main goal of these early months was the development of a new constitution for the RSI of which the first phase was to be an assembly in Verona (14 november 1943).
The words spoken at the assembly by an otherwise obscure Fulvio Balisti give a good idea of the fact that the people involved understood the almost absurd situation they found themselves - torn between old republican ideals and support for the Nazi occupants.
Yet still today all our issues are bound ... besides the need for violence, to a need for education. All our work is essential, now and tomorrow ... to fertilize the national soil ... because, if the motherland has fallen in disgrace for cowardice and betrayal, it won't rise again if not for sacrifice and valour. We live a tragedy because the problem of the Republic is a problem of freedom but the necessity of the war require restrictions.
There was to be no solving of these issues within the RSI, no victory of the war, no future for fascism. Yet, despite the odds, new issues were growing everywhere in a confuse mix ranging from anarchism to fanatic support of the Nazi, from racism to universal love. Again from Balisti's intervention:
The tree of fascism had been cut by the axe of tragedy [!]; from that stump new trees need to rise, born out of sacrifice, grown and flowering with thought and integrity...
He was not entirely wrong. The fascist regime, after suppressing and defeating opposition and dissent outside, had became proficient in suppressing dissent inside, thus throwing away much of its original energy, preventing discussion and basically binding itself to its own Procrustes' bed.
The fall of fascism had freed those energies who were suffocating inside the party. Mussolini had no longer a particular interest to stop them, given that they could not harm him in any way any more and at the same time he was aware that there was going to be no real development of those ideas.
TL;DR After 20 years Fascism was sclerotized. Its fall gave everybody on the inside the opportunity to put forward his own idea. Mussolini did no longer stop the extreme positions because mostly they were of no real consequence.
Sources: De Felice, Mussolini 1943-45 and excerpts quoted (translation is mine)
Given the huge amount of different and contrasting issues I tried to give an idea of the situation. It is hard to encompass everything. Hope you find this helpful.