Like most political issues - trivial or paramount - this one was a subject of controversy and friction within the Fascist Party. It is to be expected, as the introduction of a "racial legislation" is often referred as the choice of the Fascist Regime that appeared the least agreeable to the population (at least until Italy's participation to WW2). It is also a series of policy that the Fascist Party and related organisms tended to apply and execute effectively (save a few exceptions that were nonetheless legally regulated), and that - especially since they didn't reach the extremes of the subsequent laws of 1943-44, but remained within (already quite ample) boundaries of a sort of "segregation" model - were met with, at most, passive resistance.

It's necessary to keep in mind that by the late 1930s, while the Party did function in many ways as a coherent and rather "inclusive" or "manifold" organism, and while Mussolini's authority was essentially unquestioned (and certainly not openly challenged), the Fascist Party had never really overcome its original factionalism, frictions between currents, personal rivalries and at times violent enmities. This was at least in part a consequence of the fact that the PNF aspired to play the role of the supreme "intermediate body" between society and the more "abstract" institutions of the State; which, in practice, meant offering a "political" space to the representatives of banking and finance cartels, as well as to populist "anti-capitalist" and anti-finance instances, national-conservatives and former radicals, protectionists and liberists, etc.

Over and over, a political crisis, an economical turn, a mere shake-up provoked by internal episodes, had resulted in a readjustment of the internal balance of the Party structures favoring one group over the other, both at national level and locally. The history of the Party ranks is full of examples of this pattern: from Cesare Rossi and Aldo Finzi, to the most "infamous" rise and fall of Augusto Turati.

This to keep in mind that any controversy over the introduction of a new policy, when seen from within the Party, stems from both the Party leaders' interpretation of the ethical, social and political repercussions of the policy, and from the internal dialectics of the Party organizations themselves. And it's not exactly easy to tell which one of the two aspects weights more in each case.

As a side note, I hope I have interpreted your question correctly by focusing on Party leaders and not just on "party members" (by 1938 male organizations alone counted well over two million members, after membership had been made a requirement for public employment and career advancement in many sectors).

Now, on this specific point, we have quite a few sources that provide a picture of the "reaction" of the Party leadership to the "definitive" deliberation the upcoming "racial legislation". In Ciano's diary - for the meeting of the Grand Council of October 6^th 1938 - one finds:

Grand Council. The problem of the Jews. Balbo, De Bono and Federzoni speak for [them]. The others [De Vecchi was the only absent] against. Especially Bottai who surprises me for his intransigence. He opposes any mitigation of the measures. "They'll hate us because we drove them away. They'll despise us because we take them back."

Bottai's own version is quite longer. Mussolini had introduced, illustrated and explained the measures almost entirely by himself. Farinacci, who had spoken immediately after in support of the measure, had not made much of an impact. Starace had also vocally supported the measure (but this was to be expected as it was obvious that the issue mattered to Mussolini and could help strengthen the "moral fiber" of the Party somehow). The main opposition had come from Balbo - something which Mussolini had expected, since he had opened his exposition with a reference to an inquest by Nello Quilici on the "Jewish penetration in the political-administrative-cultural fabric of Ferrara".

The city, once Balbo's personal feud, had been for twelve years under the administration of Renzo Ravenna, a personal friend of Balbo and a member of the local Jewish community, until the latter had been recently forced to resign in March 1938 by the mounting anti-semite campaign in preparation to the introduction of the official legislation.

In Balbo's case, we can safely say that his attempts to mitigate the new legislation were inspired at least in part by a genuine distaste for the antisemitic turn of the Fascist Regime. At the same time, already in his squadrist days in Emilia, Balbo had frequently been accused of being too close to financial interests, especially to the agrarian groups that bankrolled his action squads, and thus the attacks of old "intransigents" (albeit usually on opposite sides) like Bottai and Farinacci on the "Jewish question" could be presented under a different light.

This was also the case of Federzoni, whose "lukewarm" attitude could easily be presented as expression of "conservatism".

There were, of course, significant possible repercussions for the stability of the Regime that had to be considered. From this side, besides possibly further damaging the international reputation of Italian Fascism (a factor that probably shouldn't be overestimated, as the idea was that "international Jewry" was already hostile to the Regime and therefore already damaging its reputation before the laws were introduced), there was the fairly contentious issue of mixed marriages and of the "racial" status of converts. A portion of the discussion had centered around the need to avoid the - extremely irksome for the Holy See - equiparation of marriages between a Catholic and a Jew (for the purpose of "racial" classification) to concubinage. This didn't accommodate the issue entirely (the Pope wrote directly to the King and to Mussolini about a week before the publication of the final decree) but probably helped mitigating the fallout within the Catholic world.

On the opposite end - whether present at the Grand Council or not - the Fascist leaders less sympathetic to the alignment with Nazi Germany tended to look at the antisemitic legislation as a concerning moment in a project they didn't favor (Grandi would be the obvious example); even if, as we saw, Bottai was extremely reluctant to mitigate any of the administrative measures already introduced without being overly sympathetic to the recent political developments.

These political considerations behind the antisemitic measures were often acknowledged quite directly. As Farinacci wrote to Mussolini in early August 1938 - discussing the recent publication of the "race manifesto" by a group of Italian academics and researchers:

To be frank, the racial problem, from an anthropological point of view, has never really convinced me. This is a genuinely political problem. [...] On philosophical and scientifical grounds, there's always a debate; on political grounds, where State reasons are at stake, one has to act and to win.

But Farinacci - an "intellectually reactionary" radical who had spent over a decade championing an "intransigent" interpretation of the Fascist revolution, and saw in the alignment with Nazi Germany an opportunity towards that end - had many reasons to support antisemitic laws without invoking "anthropological" motives, with the Jews representing the persistent allegiance of portions of the Italian society to foreign "bourgeois-finance" elements, in spite of the efforts of the Regime to build and consolidate a true national community. A view not much different from that of one of the most famous Italian antisemites - this one certainly more sincere in his support for the eradication of Jewishness from Italian society and culture - that Giovanni Preziosi whose association with Farinacci dated back to the 1920s, back when the latter supported the Neapolitant "dissidents" operating under Vincenzo Tecchio, of whom Preziosi had been a close collaborator.

As said before, it's difficult to tell the two sides apart.

I hope this offers at least a sketch of the general picture. There is certainly much more that could be said and, if you happen to have any follow up questions, feel free to ask. I have gotten my second dose recently and I am a bit under the weather, but I'll do my best to reply in a timely manner.