A Koine would presumably have to be ubiquitous. Levman’s Koine is only ubiquitous in its absence (as far as my grasp of facts go).
Almost every Pali Indo-Aryan word (barring the occasional loanwords from outside Indo-Aryan) has a sanskrit (Vedic or classical) i.e. Old-Indo-Aryan cognate, and the phonetic and grammatical changes therefrom that lead to canonical Pali are in general well documented. The same linguistic and non-linguistic (orthographic) processes that gave rise to standard Pali vocabulary also are broadly applicable for the orthographically attested variant word forms found in early-Jain, Ashokan etc texts. They all eventually go back to Old-Indo-Aryan word-forms directly and transparently. So a Middle-Indic Koine has no necessity to exist, and there is no evidence that it did exist.
Also it is worth mentioning that Pali and the other so-called early forms of Middle-Indic (Gandhari, Ashokan, Ardha-Magadhi) did not have any native names or regional distinctions or ethnic communities between the 4th and 2nd century BCE - thus they were not true vernaculars (in the phonetic form that they are found attested in).
From the Pali canon, it would appear that everyone across northern-India in the 4th century BCE spoke the same language and understood each other perfectly well. In the Ashokan inscriptions spread across the subcontinent, the same thing applies to the language, the language and vocabulary is verbatim the same all over in multiple copies of each edict. The language only changes in the Ashokan edicts in Afghanistan and other areas close to Afghanistan (to Greek and Aramaic). This again means the whole of northern and central India proper was predominantly speaking one language. In the early ardhamagadhi texts, the picture is the same. This picture (of a single Indo-Aryan language being spoken across northern India) is therefore a consistent one.
The early Sanskrit texts coeval to the Buddha and immediately thereafter (which are hundreds of times more numerous than all kinds of middle-Indic) actually claim that Sanskrit is the native language of Indo-Aryan India (i.e. excluding the Dravidian-speaking south) i.e. apart from also not being discordant with the other evidence from the Pali, Ashokan, Ardhamagadhi sources - they actually claim that people spoke Sanskrit (not exactly standard Paninian sanskrit but dialects close to it - of which the Paninian register was adopted as the standard). They only distinguish the Chandas (early Vedic) and the Bhāṣā (late Vedic, which includes classical Sanskrit) as being languages. After Sanskrit, we don’t see any mention of newer vernaculars until after the start of the Common Era.
Therefore, in the absence of evidence of any middle-Indic Koine, the 4th century BCE Indo-Aryan Koine is classical sanskrit itself, as I have already said here as well.