But what about before the fifth century? How far back can
pronunciation be reconstructed with any certainty?
If you are talking about individual features of pronunciation, then I think we can go quite far back, maybe even all the way to the Greek parent language. If, however, you are talking about the pronunciation of complete words and phrases, then I think our confidence decreases rapidly beginning shortly before the introduction of the Ionic alphabet in Athens.
For instance, because of the spelling, we can say that Eta and Epsilon must have represented different sounds and phonemes at one point and that Eta seems to have developed from Alpha; however, I don't think we can reliably say how Solon, for example, would have pronounced this vowel during the sixth century specifically, let alone how Homer would have pronounced it centuries before.
Is it possible to reconstruct a proper Homeric pronunciation for the
Iliad, for example? And if not, how close can we get?
We also have the problem that the "copies" of Homer's Iliad that have come to us through the tradition are "composites" of versions that passed through various sub-traditions. Most scholars assume texts such as these probably originated as a fluid oral text, then were passed down and reedited among different dialect speakers, then were written down in various forms, then were later "canonized" in particular forms, and then were edited by later Hellenistic scholars into the form that we use today according to what they thought was a cultured pronunciation consistent with their time and the spelling they inherited. Scholars analyzing the text seem virtually certain that it was edited and re-edited at various ancient stages to update it and conform to the pronunciation habits of different dialect speakers.
What we have then is an "epic" dialect that is also a composite of several dialects, all of which would have been slowly changing in pronunciation over the centuries. At different time periods, after the introduction of versions of the Greek alphabet, we can be roughly certain about various dialect pronunciations, but not about how they would have sounded centuries before.
In certain cases, we can certainly reconstruct certain sounds that some performers would have pronounced. For instance, as you probably know, some of the verses scan properly only if we assume that some of the words were pronounced with the digamma that fell out of Athenian pronunciation but was retained in other dialects down to classical times. You can certainly add this to your own pronunciation, but will have no assurances that other aspects of your pronunciation will be consistent with this addition if you otherwise use fifth century Athenian pronunciation as your guide for other sounds. In other words, since we are dealing with a mishmash of dialects, possibly from different periods, it is not possible to reliably reconstruct how any particular performer would have pronounced all the sounds before the time of the reconstructed 5th century Attic pronunciation.
As a comparison with modern English, we know that some speakers of American English still distinguish /w/ from /hw/, others distinguish all the vowels in "Mary," "merry," and "marry," and yet others merge the vowels in "cot" and "caught." Even knowing this is not enough to confidently project backward how any particular actor would have pronounced Shakespeare in a theater in Philadelphia in the early 1800s. We can, of course, give approximate dates to many sound changes, but I don't think we can map everything out so exactly in even so recent a time. This is what we face in trying to reconstruct any particular pronunciation of the Iliad much before it was supposedly canonized in the "Peisistratean recension" in the late sixth century BCE. We know that some features of older pronunciations had already been dropped, but some may also have been added to conform to Attic pronunciations.