You don't really need to begin with mass literacy - if printing takes off, you can rather end up there. Evolved cuneiform - say Akkadian - is a very complex system, but not really significantly more so than Japanese or Chinese. A movement towards mass literacy might simply lead to a greater use of syllabic writing (that is fairly esasy, I once taught myself the Assyrian syllabary, then realised what else lay in wait and gave up).
Also, we should not make the mistake of taking the prototypical Mesopotamian society - the Middle Babylonian period - as representative of Mesopotamia at all times. In the first millenium BC, urban civilisatrion in the orient underwemnt some very interesting changes, including monetisation and, indeed, a wider spread of literacy. In this atmosphere (our easiest access to its 'feel' are the prophetic books, Ecclesiastes and Proverbs, but there are many sadly inaccessible documents that bear witness to its brisk economic and social pace), printing could take off.