Walker's 1822 pronouncing dictionary says
... nothing is so vulgar and childish as to hear swivel and heaven with the e pronounced distinctly, or novel and chicken with the e suppressed.
It seems that in 1822, there were some words (chicken, aspen, patten, leaven) where the en was pronounced with a vowel—probably [ǝ]—and some words (harden, heaven, fallen, burden), where the en was pronounced as a syllabic consonant [n̩]. Today, we consider these as allophones (so they are completely interchangeable), but apparently not back then for upper-class British accents.
It is difficult to sing syllabic consonants, so presumably words like heaven were treated as one-syllable words in music, where the 'vn' was pronounced quickly between the notes (e.g., "and Heaven and nature sing" in the 1719 Christmas carol Joy to the World). All three uses of heaven in Handel's Messiah are spelled heav'n and sung on one note, for example
He that dwelleth in Heav'n shall laugh them to scorn,
however, listening to these sections on the internet, modern singers use two syllables much of the time, showing that you're not the only one who's confused about how to pronounce heav'n.
In poetry, Shakespeare treated heaven as either a one- or two-syllable word, depending on which he needed to make the lines scan.
For example, we have
Sometime too hot the eye of heaven shines,
from Sonnet XVIII, and
To shun the heaven that leads men to this hell,
from Sonnet CXXIX. To make these lines scan, heaven must be pronounced with two syllables in the first and with one syllable in the second.
Shakespeare didn't do the same thing with burthen/burden, so maybe the distinction between syllabic [n̩] and [ǝn] goes back to the 16th century (although some words like burthen would have had to change their pronunciation).
So my guess is that heav'n is pronounced with a syllabic [n̩]: [hɛvn̩]. It still seems like two syllables, but it's easier to compress into the space of one syllable than [hɛvǝn]. If you know French, think of words like livre /livʁ/ and table /tabl/ where the /r/ and /l/ are kind of tacked onto the end of the syllable.