I have no definitive sources or knowledge of anyone who has studied this specifically (though I’m sure someone has), so what follows is just my own thoughts and deductions, paired with my own experience of how certain sequences and words tend to be pronounced.
I don’t know for sure whether I’m right, but a simple explanation based in general articulatory principles seems the most economical and easily reproducible candidate.
Well… I say “simple”. The explanation itself is really simple enough, but it requires quite a lot of knowledge about phonetics and articulation, so the answer is dreadfully long. See at the end for a tl;dr.
On pronouncing /r/
The sound /r/ is fairly unique in the English language in being the only consonant that requires retracting the tip of the tongue, and in most dialects also to some extent retroflexing it; that is, moving the tip of the tongue upwards towards the hard palate (or alveolar ridge) while keeping the dorsum (the mid body of the tongue) low. At the same time, it’s labialised (pronounced with rounded lips) in most dialects.
The two most common realisations of the phoneme /r/ are as a postalveolar approximant [ɹ̱ ~ ɹ̱ʷ] or a retroflex approximant [ɻ ~ ɻʷ] (for simplicity and legibility—and because I can type it without having to hunt down the symbol in a character map—I’ll just use [ɹ] here in broad phonetic transcription, even though that’s actually a slightly different sound). Both of those, and most other variants, have in common that they require moving all three articulatory parts of the tongue: the tip/blade (upwards), the dorsum (downwards), and the root (backwards). Most other consonants only require moving one or two, letting the others simply go along with whatever movement requires the least effort: if the tip of the tongue moves upwards towards a /t/, the dorsum moves up a little bit as well, and the root doesn’t have to move at all, for example.
So in several ways, /r/ is articulatorily more complex than other consonants.
In addition, /r/ is essentially a glide and phonetically really more vowel-like than consonant-like: there is very little blockage of air flow, and it is often difficult to tell exactly when a surrounding vowel stops and the /r/ starts. This is why we get ‘r-coloured’ vowels, particularly in rhotic dialects where postvocalic /r/ is always pronounced. Essentially, because the tongue movement is so complex, it takes a bit longer to get to and from the proper position; and because there is no blockage of airflow, what we perceive as an /r/ ends up taking over some of the duration of the vowel. These two factors mean that /r/ ends up being a bit longer than most other consonants.
Reducing tongue movement
In words like prerogative and Prahran, we have a sequence of unstressed /rɪ̵r/ (/ɪ̵/ is a phoneme whose realisation vacillates with various degrees of randomness between /ə/ and /ɪ/), in which the tongue has to move quite rapidly into /r/ position, then back out to a neutral vowel position, and then back into an /r/ position—without ever touching any part of the mouth to break the ‘flow’. It’s not so strange that in rapid speech, this fairly complex movement gets simplified somewhat.
The easiest way to do this is to reduce the movement back to neutral position for /ɪ̵/, by only letting your tongue get halfway towards it. And then, once you’ve got that, you can really just leave the tongue more or less where it is. After all, there aren’t any parts of the mouth that the tongue has to touch, so there aren’t any really obvious giveaways if you don’t move your tongue at all. Keeping your tongue in place, but allowing the corresponding amount of time as you would have used for an /ɪ̵/ to pass, you end up with a syllabic /r/, i.e., an approximant/glide that has the same intensity, voicedness, and length as a corresponding pure vowel would: [ɹ̩]. This sound is perceptually very, very similar to [ɚ], the r-coloured schwa found in American English better [ˈbεɾɚ]. In fact, depending on just how much of the vowel the /r/ is allowed to take over, it can be almost completely identical to it.
So if you take prerogative /prɪ̵ˈrɒɡətɪv/ and reduce the tongue movement required get into position for the /ɪ̵/ to basically nothing, you end up with [pʰɹ̩ˈɹɒɡətɪv], which I would argue is all but indistinguishable from [pʰɚˈɹɒɡətɪv]. Given how /r/ has a tendency to encroach upon preceding vowels, it can even easily be perceptually quite indistinguishable from [pʰəˈɹɒɡətɪv]: we’re so used to /r/ colouring a preceding vowel that find it difficult to distinguish between coloured and non-coloured vowels before it: how much difference is there really between the first bit of prerogative and peremptory? To me, they’re all but identical except in very carefully enunciated speech.
Going one step further, it is even possible to reduce the sequence [ɹ̩ɹ] to a single [ɹ], completely losing the syllabic status of the approximant—that is, getting rid of the first syllable entirely, yielding [ˈpʰɹɒɡətɪv]. This is characteristic of informal, colloquial, rapid speech, but it is a possibility. This is also possible with /ər/, as in parade or peremptory, so in this variant, there is definitely no difference between /ər/ and /rər/.
Contrary prerogatives in Prahran libraries in February
There aren’t many words where sequences of this type occur in English, but there are some. Searching a little in the CMU Pronouncing Dictionary (as per nohat’s answer here), I find at least the following (limiting myself to base words—i.e., not including derivations—and to words with any kind of actual currency in English—for instance, I’m not counting Sundararajan which, for whatever reason, is in there):
Ararat /ˈaɹəɹat/
arbitrary /ˈɑːrbɪ̵trəri/
contrary /ˈkɒntrəri/
library /ˈlaɪbrəri/
prerogative /prɪ̵ˈrɒɡətɪv/
Priroda /prɪ̵ˈroʊdə/
– to which we can possibly add your Prahran if we assume that Wiktionary is slightly inaccurate in its pronunciation key. Since I’ve never heard anyone say the name, I will defer to your judgment here, but for the word to fit, it ought to be /prəˈran/, rather than /pεˈran/ as Wiktionary claims, and the surface form would then presumably be [pʰɹ̩ˈran ~ pʰɚˈran ~ pʰəˈran]. I’m going to assume that for the purposes of this answer.
Another very similar word is February /ˈfεbrʊəri/, which is the only word in the English language to contain unstressed /rʊər/ and gets all sorts of mangled. A very common mangling here is replacing the first /r/ with a /j/, yielding [ˈfεbjʊ(ə)ri]; we can leave that mangling out entirely here. Another very common tactic is reducing /rʊər/ to just /rər/, which makes it a perfect candidate for our purposes.
I also think we have to leave out Ararat: since it has vowels on either side of both /r/’s, there is no way for a syllabic [ɹ̩] or an r-coloured [ɚ] to appear. An /r/ between two vowels will never be reduced in this way. So we’re left with arbitrary, contrary, February, library, Prahran, prerogative, and Priroda.
These words can be put into two subgroups: one where the /rər/ sequence comes after a labial consonant, and one where doesn’t:
After labial:
February, library, Prahran, prerogative, Priroda
Not after labial:
arbitrary, contrary
These two groups behave slightly differently when it comes reducing the /rər/ sequence:
Both groups can be pronounced according to their phonemic structure, with a fully enounced [ɹəɹ ~ ɹɪɹ], but I’d say you’re not very likely to hear it in natural speech.
Both groups can also undergo complete deletion of the vowel/syllable, yielding just [ɹ]. At least, I assume Wiktionary’s mention of Prahran being colloquially known as ‘Pran’ means that [pʰɹan] is a possibility there. The rarity of the name Priroda and the fact that it’s a foreign name would probably make this a very unlikely pronunciation for that word, but if in context it is a well-known, common word, it should be a possibility. For all the others, colloquially, you are quite likely to hear [ˈɑːbɪtɹi, ˈkɒntɹi, ˈfεbɹi, ˈlaɪbri, ˈpɹɒɡətɪv].
But strangely enough, the ‘intermediate forms’ with [ɹ̩ɹ ~ ɚɹ ~ əɹ] (I’ll just write [əɹ] here) only seem to be possible for the labial group. You’re quite likely to hear [ˈfεbəɹi, ˈlaɪbəɹi, pʰəˈɹan, ˈpʰəɹɒɡətɪv, ˈpʰɪɹoʊdə] in casual speech, but [ˈɑːbɪtəri, ˈkɒntəri] both sound distinctly odd to me.
Whether labials are the only deciding factor here, I don’t know. The sample set is too small. I would guess that stress also plays a part:
The intermediate forms seem more likely to me when the /rər/ sequence is pretonic (before the main stress) than when it’s posttonic (after the main stress): [pʰəˈɹan, pʰəˈɹɒɡətɪv, pʰɪˈɹoʊdə] seem more likely to occur than [ˈfεbəri, ˈlaɪbəri].
Conversely, the elided forms seem more likely to occur posttonically: [ˈfεbri, ˈlaɪbri] sound more common to me than [ˈpʰɹɒɡətɪv] (and I can’t really speak for Prahran and Priroda).
There are probably more words that fit the pattern as well, and they may shed more light on the factors that influence these reductions; but the rather arcane phonetic respelling system used by the CMU Pronounce Dictionary makes it fairly difficult to find them.
tl;dr
Because of the articulatory properties of /r/, it tends of eat away at, and to some extent merge with, a previous vowel. It does so because we’re lazy and want to reduce the amount of movement our tongue has to do; that way, we can communicate faster. When the preceding vowel is a neutral /ə/ or /ɪ/, the resultant merger is more or less identical with a simple syllabic /r̩/.
Words like prerogative have unstressed sequences /rər/, which can be reduced to /r̩r/, /ər/, or even just /r/ because of this tendency of /r/ to ‘eat vowels’.
There is a difference in whether the consonant preceding /rər/ is labial or not: if it isn’t, the variant /ər/ seems to be blocked. Stress also plays a part.
