With thanks to this discussion for pointing me in the right direction, I found a passage in W. Sidney Allen's Vox Graeca, pg. 52-53 that addresses this question:
In compound words, however, one has to consider the possibility of aspiration of the second member, thereby giving rise to medial aspiration, or 'interaspiration' as it is commonly called. When the first member ends in a voiceless plosive, this is of course an established fact (e.g. ἐφορῶ from ἐπ(ι)+ὁρῶ), the aspiration having become a feature of the plosive. But the situation is less clear when the first element ends in a vowel or in a consonant which has no aspirated counterpart.
The rough breathing marking is "occasionally" inserted in Attic inscriptions, though usually not:
In such cases the aspirate does not generally appear in Attic inscriptions which otherwise indicate it, but it is occasionally found--e.g. ευhορκον, παρhεδροι, προσhεκετο (= προσηκέτω).
As mentioned, Latin transcriptions sometimes show it and sometimes do not.
Latin transcriptions show considerable variation, and this may have been a feature of Greek speech itself; the presence of aspiration in such forms could well have depended upon the extent to which the two elements of the compound were still recognized as such by the speaker.
Explicit interaspiration is present in some words:
Apart from compounds (and exclamations as εὐαἵ), interaspiration is attested for Attic only in the word ταὧς 'peacock', a borrowing of uncertain origin, which is specifically mentioned by Athenaeus.
I tracked down this last citation, which is mentioned in the LSJ entry:
ταὧς λέγουσιν Ἀθηναῖοι τὴν τελευταίαν συλλαβὴν περισπῶντες καὶ δασύνοντες (Tryphoap. Ath. 9.397e)
I think the key point is the one I bolded above: given the haphazard inclusion of rough breathing in written testimonies (both Greek and Latin), it is likely that there was no consistent convention for pronouncing these words.