According to the Oxford English Dictionary (OED), the word blue, a loanword from "Anglo-Norman blew, bliu, blu, blwe, bluw", had a number of spellings that were used during the Middle English period, among them <belewe>:
eME bluȝ, ME bleu, ME bleuh, ME blu, ME bluwe, ME blw, ME blyu, ME blyw, ME–15 blwe, ME–16 blewe, ME–17 blew, ME– blue, 15 belewe, 15 bliew, 15 bliewe; Sc. pre-17 bleu, pre-17 blewe, pre-17 17– blue, pre-17 19– bew, pre-17 19– blew, 19– blyew, 19– blyue, 19– byoo; N.E.D. (1887) also records a form ME bluw.
The Wikipedia article about the term "blue moon" cites some lines of verse that it says contain the earliest known example of blue moon (although actually, the phrase does not appear in exactly that form in the document), from a pamphlet "Rede me and be nott wrothe, for I say no thynge but trothe" that was published in 1528:
Yf they saye the mone is belewe /
We must beleve that it is true
(These lines also appear in the OED entry on moon as the earliest and only quotation for the now-obsolete expression "†to say that the moon is blue", which the OED compares to the still-current "to believe that the moon is made of green cheese".)
The vowel in the stressed syllable was a diphthong something like [iu]
The rhyme constitutes evidence that <belewe> here was pronounced the same from the stressed vowel to the end of the word as the word <true>. The OED entry for the word true has some more details about the pronunciation of that word in Middle English:
The α. forms show the usual development of Old English ēow to the Middle English diphthong ēu, ēw (with long close ē). By the 15th cent. this diphthong had fallen together (perhaps as /ɪu/) with the reflex of Old French u (i.e. /y/), giving rise to the Middle English and later β. forms tru, true, etc.
The symbol [ɪu] represents a diphthong that is not found in most common English accents, but that consists of a vowel like the "ih" in the modern English word "kit" followed by an "oo"- or "w"-like offglide, like at the end of the /aʊ/ dipthong in the modern English word "cow". I have also seen this diphthong transcribed as iu, ɪʊ, and iʊ ("Diphthongs", English Language and Linguistics Online). The IPA symbols ɪ and i and ʊ and u refer to theoretically adjacent and in practice actually overlapping areas of the "vowel space", so the variation in transcription mentioned here doesn't actually correspond to much if any difference in the sounds that are being referred to.
The consonants /b/ and /l/ were pronounced about the same in Middle English as they are today.
I am more uncertain about the pronunciation of the first and last <e>s. In Middle English texts, the letter <e> could represent a number of sounds: an unreduced short vowel /e/, one of the long vowels /eː/ or /ɛː/, the schwa /ə/, or no sound at all (a "silent e" as in many modern English words).
I expect that a lot of material has been written about the topic of when word-final schwa was lost in English, but unfortunately, I haven't read almost any of it, so the next bit of my answer is based on only a single source: "A U-turn and its consequences for the history of final schwa in English", by Donka Minkova (2015). But it seems that the general relevant trend of word-final schwa loss is fairly uncontroversial. Minkova says:
there is no scholarly disagreement on the full-blown progress of final schwa loss from the 12th century onwards. Not all contexts were equally conducive to reduction. Inflectional final vowels are more prone to reduction and apocope. There are significant dialectal differences: the loss of schwa in word-final position, both inflectional and stem-final, was more advanced in the northern dialect areas, where by the mid 14th-century its realization would have been an archaism.  The insertion of unetymological <-e> is a good test for the instability of base-final <-e>. The Ormulum, an autograph manuscript dated c. 1180, written in Lincolnshire, shows a number of nominatives in <-e> in nouns which had no final <-e> in OE: axe < OE æx ‘axe’, blisse < OE bliss, wunde < OE wund ‘wound’.  In the southern dialects apocope spread more slowly, with inflectional <-e>’s leading the way. Chaucer, whose reconstructed pronunciation shows other conservative features, resorts to final schwa for metrical purposes, but his usage is more constrained than the ubiquitous /-ə/# of early ME: for nouns it is consistent only in monosyllables in prepositional phrases. The dialectal details are much discussed in the literature (see Minkova 1991: Ch. 2; Fulk 2012: § 30). Of interest in the context of the overall history of the change is that by the beginning of the 15th c. the realization of schwa in word-final position in the base forms of nouns in all dialects had become obsolete. This allows us to posit an active constraint on the distribution of /-ə/ as defined in (3):
(3) */-ə/#: Avoid the realization of schwa in word-final position
Based on this information, it seems that the final <e> in <belewe> was most likely silent and did not constitute a distinct syllable, since the passage it appears in seems to have been written in the 1500s, not only long after the 12th century but well past Chaucer's time also.
The first <e> is puzzling, since it is not consistent with the etymological origins of the word. It seems possible that it was an inserted unetymological silent <e> letter, somewhat like the examples given by Minkova, although it does not occur in word-final position like these examples. If the first <e> did somehow correspond to some actual vowel sound, I would guess that the schwa would be most likely.
Summary of pronunciation and syllabification
It seems to me that the pronunciation of the word <belewe> in this passage was most likely monosyllabic /ˈbliu/, so I wouldn't recommend trying to break the word into syllables at all. But if you did, I think you'd definitely want to avoid dividing it as bel·ewe. It seems pretty clear that the /l/ fell in the same syllable as the following diphthong /iu/: even if there was some kind of vowel preceding it, that vowel would have almost certainly been unstressed, based on the word's etymology and the rhyme with true. E.g. things like /ˈbel.iu/ or /ˈbeː.liu/ seem completely impossible to me; and while I'm not confident enough to completely rule out /bəˈliu/,
(whose syllables would be divided as be·lewe), it seems less probable to me than /ˈbliu/ belewe.
That's my best shot. I'm sorry, I'm clearly a fair bit of my depth here: I only know enough about Middle English to give an answer based on general information about Middle English sound systems, not an answer that addresses which variant pronunciations were most likely based on factors like time, place, dialect, and choice of spelling variants.
For a detailed, educated answer that talks about things like what the unusual spelling of blue in this particular document, with <bel> instead of <bl>, is most likely to have corresponded to in terms of pronunciation, you may have to get in touch with professors or other kinds of experts who have seriously studied Middle English phonology and paleography.
A suprising hypothesis about the etymology of "blue moon" that is mentioned in the linked Wikipedia article prompted me to post the following, separate question: Is the "blue" in "blue moon" a reference to betrayal?