Historians have asked themselves this for centuries, and it's hard to tell, not a lot of sources give us that perspective. Most relay the story but don't comment on it, except for Livy, who is skeptical of the whole "raised by a she-wolf" stuff (but has no problem with the "Mars laying with Rhea Silvia" part of the story). But I think there's a more interesting question behind this one, and solving that one can help us answer yours. Bear with me, please.

Do we know if the myth contained these elements from the earliest times of the Kingdom of Rome? Themes of fratricide, rape, and of the original Romans being a community of outcasts and bandits? When did these themes enter the story, and why?

We do know Romulus was an early figure in Roman consciousness, while Aeneas seems to be a later adoption to the story, mostly a Greek concept (even if there may be traces of an Aeneas-centered hero cult in Latium, but it's disputed). Sometimes Aeneas founded Rome (as stated by Hellanicus of Lesbos), sometimes Rhomos, son or grandson of Aeneas did, sometimes it was Evander, traditionaly considered the Greek who brought Greek customs and religion to Italy (Fabius Pictor), and sometimes Odysseus and Aeneas did it together (so said Dionysius of Halicarnassus in an small fragment discussed by F. Solmsen). The story of the twin founders goes way back, but did the fratricide? If so, how was it interpreted? A cautionary tale? A show of strength? A demonstration of the ruthlessness that would characterise Late Republican and Early Imperial Roman politics and expansion?

Rodriguez-Mayorgas says that most of the elements of the story were already present at the times of the Punic Wars, and respond to the same intent as Manetho in Egypt or Berosso in Babylon: Greek-style history of important kingdoms, but brushing off the excessive hellenic influence, especially considering there was a lot of pro-Carthaginian history being written in Greece, much of which is now lost. For more on that, see: https://www.academia.edu/758835/Romulus_Aeneas_and_the_Cultural_Memory_of_the_Roman_Republic

Now we more or less know that Romulus was there from the start, but the murder? And, more importantly, the murdered?

T.P. Wiseman asked the relevant question here: Why Remus? Romulus founded Rome, it was a common, established practice in Ancient times, to have a god, hero, giant or supernatural entity found a city. But why murder his twin brother? Is this an explanation of the two consuls? A fossile of the Roman-Sabine dual foundation? An echo of the now famous Indo-European twin sacrificing twin story?

In "Remus: A Roman Myth", Wiseman says that Remus evolved as a representation of the Plebs (hence his founding of the Aventine, the hill traditionally associated with the poor) following the violent clashes between Patricians and Plebeans of the 5th and 4th Century BC. At that time, fratricide was the way some understood Rome's internal struggles (metaphorically and sometimes literally), and therefore myths were developed and solidified into theatre plays and musical rhimes. In some, Remus was a hero sacrificing himself for the plebs; in others, Remus was the rebellious fool who is felled because of his hybris.

This last version is the one that caught on, as conveyed by Patricial writers. Once the immediate context of Patricio-Plebean conflict vanished, the notion of Remus as the "father of the Plebs" also went away, and he was left as a fool who rebelled, while Romulus was the righteous hand of the father punishing his wanton child. Roman histories of the Republican period are chock-full of parents who kill their children, wives who kill themselves before being dishonored, generally speaking parent figures whose first duty is to virtue and to Rome.

So, to give a kind of straight answer: Mid to Late Republican Romans believed that Romulus did a regrettable thing (fratricide) for a good reason (varied reasons are given, sometimes it's punishing hybris, sometimes it's a necessary sacrifice to ensure the greatness of Rome, sometimes it's the maintaining of order), and therefore he was revered for having put his duty above his feelings for his brother, the stern but necessary reaction a father figure must understake against disorder, both in the house and in the State. Thus, Romulus had the dignitas befitting a patrician and a leader of the city. Remus was a troublemaker.

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By the way, there's other variations on the Romulus and Remus story that were written after the Aeneid, like the Origo Gentis Romanae (check it out here http://www.roman-emperors.org/origogentis.pdf) , in which the murder doesn't happen and Remus founds the city of Remuria or Lemuria nearby. Just to show that maybe the story of the fratricide not only wasn't totally prevalent, but also that if it had been so during late Republican or early Imperial times, it eventually subsided, and more pallatable foundation stories developed. With less fratricide.

I know I haven't answered your question straight-on, but I don't think there are sources for a satisfactory answer, and the tangent is relevant to the question.

This may not be quite what you're looking for, but I've got a writeup here about the story of Aeneas, and how that narrative (of an exiled Trojan hero escaping to Italy and founding Rome or an ancestor-city) evolved and gained importance over time.

One important point to note is how it meshes with the Romulus-Remus narrative - namely, that it doesn't, but that it was also clearly too popular and significant to simply replace. So when the poet Virgil comes to write his great epic of Aeneas' story, he has to fudge things considerably to make Aeneas the 'father of the Roman people' and ancestor of Romulus and Remus, which then ends up fitting the two stories together rather awkwardly.

The key takeaway, as others have touched on, is that foundation stories are conscious decisions, at some level - people may have them passed down from previous generations, but then have to choose to continue telling them, or replace them, or remove uncomfortable details, or add embellishments. So there's always something meaningful in the stories that they do end up telling. Mary Beard (recently in her book SPQR) does a good turn on how the Romulus-Remus story, for instance, comes to reflect later (chiefly, turn-of-the-millennium) Roman anxieties around their incessant civil wars, their cultural preference for masculine, rough-and-ready virtues, and a certain sense of being newcomers and even interlopers in a much more 'civilised' (Eastern) Mediterranean world.