how should δʼ ὅτε be pronounced?
My guess – and it’s just a guess – is that δʼ would have been pronounced without aspiration, even in 5th century Athens. But this is a fraught question: do you mean the original pronunciation of the poems when they were first reduced to writing, hot off the press, presumably sometime before 500 BCE, or the pronunciation of classical Athens in the 5th and 4th centuries BCE, or even later pronunciations?
It’s important to keep in mind that the diacritical marks – breathings and accents – were added to the text long after “Homer”, beginning in the Hellenistic period. By that time spoken Greek was universally psilotic (the rough breathing had disappeared from speech).
The breathings (and accents) were added by scholars in accordance with 5th century Athenian pronunciation, which was not yet psilotic (at least in upper-class educated speech), based on a continuous tradition stemming from the period of classical Attic. (This is also true of other non-Attic texts.) But with respect to psilosis, classical Attic was conservative in relation to other Greek dialects, and there is evidence that the original Homeric language was already psilotic, as Ionian speech wold have been by the 5th century.
For example, while the word ἡμέρη is marked with a rough breathing in the Homeric texts, in accordance with Attic pronunciation of ἡμέρα, the etymologically equivalent and more prevalent Homeric word ἦμαρ is not, and we find ἐπ᾽ ἤματι in e.g. Od. 12.105, indicating that this word was not pronounced with rough breathing by the time the Homeric texts were reduced to writing.
However, the fact that dh is not a Greek phoneme (unlike θ) makes me think that despite the rough breathing added much later, Greeks in all periods would have pronounced δʼ ὅτε without aspiration. But who knows? Perhaps at some point scholars insisted on a hypercorrect pronunciation with aspiration.
Your syllabification looks right to me. τρ and κλ, when τ and κ close syllables with short vowels and hence make them “heavy”/“long”, should be split between the preceding and the next syllable.
