This seems like the perfect question for David Crystal (the esteemed pioneer of the Shakespeare "Original Pronunciation")

He wrote a magnificent volume by the name of "The Oxford Dictionary of Original Shakespearean Pronunciation" in 2016, in which he gives a detailed catalog of the distinct and numerous archaic spellings and pronunciations for each word during the Jacobean period (with the printing of the folios and quarto editions), and the words that they rhyme with according to each play.

I recall finding a similar incident with the word "again":

Now Ciuill wounds are stopp'd, Peace liues agen;/ That she may long liue heere, God say, Amen.

Richard III, Act 5, Scene 5. Folio spelling.

1.WHen shall we three meet againe?/ In Thunder, Lightning, or in Raine?

Macbeth, Act I, Scene I. Folio spelling.

As seen here, and as reinforced and verified by the work of Crystal in the aforementioned work, the word "again" was pronounced as "again" (like "grain") and "agen" (as in "ten"). If "again" was pronounced two ways, it would be no astonishment if "eye" was likewise.

Here is what the beforehand mentioned "Oxford Dictionary of..." (by DAVID CRYSTAL) puts for the pronunciation of the word "eye": for A Midsummer Night's Dream see "MND"

P.S. It seems the rhyming of "eye" and words with "ie", "y" endings (such as quality, company; after your given examples) was continued into the 18th and maybe even 19th century, (at least 1794) with Blake's immortal "Tyger, Tyger burning bright" (The Tyger) poem:

What immortal hand or eye, / Could frame thy fearful symmetry?

Also, it is found an "innumerable" amount of times in the Folio and also any 16th to mid-17th century text, words which in Our modern spelling end with a "y" syllable, were spelled rather with an "ie" (and as well, sometimes with a "y"--see Lady Macbeth's "out, out damned spot" spelling in the folio where the word "fie" is spelled "fie" and then "fye" in the same line)--(see examples like: "tragedie", "comedie", "fithie", "companie") Words were spelled the way they sounded.