Very lovely to find this discussion a couple years after the fact!
I have often discussed this topic with others, so I appreciated seeing all the points of view presented. I used to be of the mind that the most appropriate choice was just deaspiration for δ’ ὁ [do], and I had never found the ουθεις for ουδεις in Threatte to be particularly convincing since, if Classical Attic lacked a /dʰ/ phonemically, it would make the most sense to render δ’ ὁ unaspirated voiced [do], and certainly not [tʰo].
But my mind has been slowly changed by the best Attic speakers, who, in addition to immaculate pronunciation, advocate aspirating the voiced stops. I used to politely oppose their point of view, but recognizing further evidence in Latin has persuaded me otherwise.
You may know that Latin authors regularly demonstrated internal aspiration in transcriptions: Pyrrhus is one example (though this is actually a voiceless R [r̥] it shows the Roman desire to represent the contemporary Greek sound), and Polyhymnia for Πολύμνια or, indeed, Πολυύμνια is especially demonstrative that, while usual Ancient Greek orthography does not show internal aspiration, it is in fact there even in the 1st century BC Attic the Romans heard and sought to represent in Latin orthography. I used to believe this was mere slavish orthographic imitation, but the regularity of the practice convinces me that this is Greek as the Romans actually heard it (at least in contemporary Athens), and is not a mere etymological spelling.
Something worth considering also, is that there is more than one glottal fricative [h], the voiceless variety, but also the voiced [ɦ]. This is a very likely candidate for the H in words like mihi in Latin: [miɦi], since its voicing permits a smooth transition to mī as a common alternative. The voiced glottal fricative [ɦ] actually occurs quite a lot in spoken English, often in how people say "ahah," especially when not especially excited in the saying.
The verb συνί̄ημι is a good place to use the voiced glottal fricative: rather than [sy.n̥ʰǐː.ɛː.mi] (note that the [n] has been rendered voiceless to [n̥] by necessity since the voiceless [h] is in the superscript), I recommend [sy.nʱǐː.ɛː.mi]. Since it's hard to see, the two variant characters in IPA are [n̥ʰ] vs. [nʱ]. I recommend [sy.nʱǐː.ɛː.mi] (with the
ɦ) as nasals tend to voice 'weaker' sounds when they are able, such as σμῑκρός being [z̠miː.krós].
Since this is a reasonable sandhi, it stands to reason that a
voiced glottal fricative could be added to a delta with no issues: [dʱ]. Indeed, [dʱ] is IPA for the consonant sound in Sanskrit and Hindi, not [dʰ], which would be impossible since choosing the voiceless glottal fricative would devoice the [d] to [t].
Given how readily Classical Attic retains aspiration where it is orthographically possible, such as καὶ οἱ > χοι, I think the best recommendation is to keep aspiration (voiceless or voiced glottal fricative) everywhere, even where the orthography cannot show it, thus δ’ ὁ = [dʱ]. In this recent episode of my podcast I attempt just that:
https://youtu.be/Wc6enX_Wo5k
To that end, I also now write συνἵ̄ημι etc.