Terminologically, I think you are interested in the number of "distinct syllables" in a language. "Syllabic phoneme" means, approximately, "vowels", but also syllabic consonant (as exist in some languages), and with the provision that the sound has to be contrastive and not allophonic (for instance, you would not count with [ə] and [ʌ] in English since the distribution is rule-governed).
This is not information generally available for languages, and Barker's underlying paper on English shows you why. For some languages it's a simply matter of computing and "trivial" searching, but even then, non-triviality lurks. Hawaiian has 8 consonants and 5, 10 or 25 vowels, depending on how you treat long vowels and diphthongs. If we take the smallest numbers, syllables are of the type V and CV, therefore there are 5+40 theoretically possible syllables. Then you would just search a dictionary to see if they all exist (there is a gap for [wū] and only 2 examples of [wu]). Taking the largest number, you get 25+200 (possibly minus 1 or 2). The count in English is much higher because we have more phonemes, and the possibilities for combination are greater = (C(C(C)))V(V)((((C)C)C)C). But not every C can appear in every positions: plus, as noted by Barker and every every other linguist, it's non-obvious where the syllable boundaries are in English, so you have to decide if [btʃɪk] is a syllable of English, given "Dabchick" (I would say no but I'm not here to argue with his algorithm). English is rife with syllabification restrictions (syllables can't start with [ŋ] unless you syllabify intervocalic [ŋ] as in the onset).
This person claims that Vietnamese has 17,974 syllables, but this is a product of combinatoric calculations, and it is noted that about half of them don't actually exist. That's a pretty high percentage, suggesting that there are non-accidental gaps. For instance, there is no syllable in English that begins with "bn", which linguists generally consider to be the result of a rules. There are no syllables composed of sCVC where the two consonants are the same and the vowel is short, with a tiny number of counterexamples like "stet", "stat": some linguists (Clements & Keyser, among others) have that as a systematic fact of English. There are no syllables having [bl] as the onset, [ɪ] as the nucleus, and a coronal non-sibilant as the coda (t,d,n,tʃ,dʒ): nobody has previously noticed that gap and proposed a rule. We don't know if that is just a gap in attestation, or an actual rule-governed gap.
In principle if you could get a comprehensive list of all words, including inflectional forms, for a language and could decisively parse words into syllables, you could count the number of actually attested syllables in a language. Most comprehensive word lists don't include inflected forms, and they are usually based on written forms of major national languages. Polish, for example, allows long sequences of consonants, but this extra combinatoric power is concentrated in word beginnings and word ends. In the context VC*V, it's controversial where to put the syllable break (this is the general problem of syllabification algorithms and the question of what to count).
In other words, no, not generally, there aren't many such enumerations.