I am not sure if the question concerns Celestine Bittle's position specifically, or this sort of argument generally. Heidegger famously mused over the question "why is there something rather than nothing?". His answer is characteristically cryptic, "it is only because we choose being-there", but he is coming from the same place as all existentialists: the nature of being is deeper than essences because existence precedes essence. So one can argue that the true contrast between being and non-being is at best obscured by distinctions and essences, it is brought forth by life, choice and action, and therefore does not entail intelligibility, although it does not have to exclude it either.
However, even if we wish to uphold the essence of being for the sake of the distinction between being and non-being and intelligibility I do not see how this gets us to the principle of sufficient reason. Let's say in the case of being sufficient reason is demanded by intelligibility. This strongly resembles Kant's transcendental arguments from possibility of knowledge to the content of phenomena, but as Kant pained to point out this only gets us phenomenal reasons, not reasons for being in itself. But even if we grant this transcendent step how would existence of sufficient reason in one very special case get us to the principle that postulates existence of such a reason for everything, always, and everywhere?
If one feels strongly enough about intelligibility of being and blurring the lines with non-being one can easily grant sufficient reason for that case, and perhaps in many other cases, but go on to reject the general principle because they feel equally strongly about, say, moral responsibility and freedom of the will. Indeed, this is exactly how Kant proceeded by restricting causality to phenomena, and those were his major motivations.