By definition, a syllabary has separate glyphs for each possible syllable (and usually unrelated or at least not systematically related ones for similar syllables, unlike alphasyllabaries). This obviously clashes in practicality with having complex syllable structure, leading to a large amount of possible syllables and therefore a large amount of symbols. How can this conflict be dealt with?
Linear B is an interesting example for this:
This system was apparently designed for a non-Greek language, as it did not fit the sounds of Greek very well. In fact, it is likely that Linear A was used to write the pre-Greek language of Crete, and the incoming Greeks adopted this writing system for their own use, but without changing how the system fundamentally works. In doing so, they developed "spelling conventions" to represent sound patterns found in Greek but not in the syllabary.
First, there are many Greek sounds that are missing in Linear B signs, such as [g], [kh], [gw], [b], [ph], [th], and [l]. To solve this problem, signs for similar sounds are used instead: p-signs are used for [p], [b], and [ph]; k-signs are used for [k], [g], and [kh]; t-signs are used for [t] and [th]; q-signs are used for [kw] and [gw]; and r-signs are used for [r] and [l]. However, while this convention was likely easily understood by ancient Mycenaean scribes, it took modern scholars a lot of theoretical analysis and work, plus comparison with later Greek dialects and reconstructed Mycenaean words to rediscover how this system works
And more relevantly:
Another inadequacy stems from the fact that Linear B signs usually represent Consonant-Vowel (CV) syllables, but the syllabic structure of Greek allows initial consonant clusters, ending consonants, and dipthongs. In the case of a syllable with a initial consonant cluster, individual consonants in the cluster are written by a CV sign whose vowel matches the vowel of the syllable. Therefore, for example, the word tri is written as ti-ri, and khrusos as ku-ru-so. In the case of ending consonant, the situation becomes more complicated. Ending consonants such as [l], [m], [n], [r], and [s] are not usually written, whereas other consonants such as [k] and [p] are written in a way similar to initial consonants.
So one way to represent consonant clusters in a syllabary is to pick a "dummy vowel" that gets read over when the word is pronounced, as in Linear B, or Japanese katakana when writing foreign words.
The question doesn't mention just how complex the syllable structure is; if it's not too complex, you can also do what Japanese hiragana does with the ん character and have a single character that represents a syllable-ending consonant. Or something like what Brahmi does for vowels and nasals, adding strokes and dots to represent them. Maybe if kra, tra, and pra are common syllables in your language, you could have a stroke that represents an r between the consonant and vowel, and add it to the characters for ka, ta, and pa.
The best solution to this is to not make your writing system syllabic if your language does not support the syllabic structure by having a low number of syllables.
I guess that one of the most commonly cited examples of a syllabic writing system is the Japanese katakana/hiragana system. Japanese phonology fits these systems very well because syllables are basically (C(y))V with the principal exception of syllabic n and the secondary one of ‘double consonants’ (まって — matte). Therefore, Japanese needs only around 50 distinct symbols for its syllabary. It made perfect sense to simplify complex Chinese characters to a reduced form carrying just a phonetic information.
On the other hand, languages such as German or English would be terrible choices for inventing a syllabary. Their syllable structures have very little constraints at all; basically anything is possible. Coming up with a syllabary would require many hundreds if not thousands of characters — to the point where it would be simpler and more consistent to create an ideographic writing system instead.
Take for example to write. In a syllabary, the symbol for write in I write would be the same as in right (there) or you are right. But then if you take I wrote, the wrote would be something completely different (yet with the same symbol as rote memorisation, I think?). I have written would use two symbols, neither of which has anything to do with the previous ones.
In an alphabetic or alphasyllabic system, there would at least be some visual resemblence between all those forms allowing for recognition. That’s also why English benefits from its extremely irregular (based on phonetics) spelling: meanings can be recognised based on writing and words with similar meanings are written similarly. Their common root is retained in a common spelling which is not possible in a pure syllabary.
A language’s writing system will either have been developed by the language itself or have been copied and adapted from geographically close languages. Of course, it would be possible for a Germanic-type language with a highly variable syllable structure to adapt a syllabic writing system initially — but it is very highly likely that it will soon be improved in numerous ways until an actually working alternative has been found.
Instead of thinking how you can bring the two ends together, I would propose you think of which writing system actually makes sense for your language.
Both the Brahmic scripts and Canadian Aboriginal syllabics have a grapheme for "suppress this sign's inherent vowel".
This could get tedious with big clusters; also, if you're starting with a pure syllabary, you have to decide which vowel to suppress!