One must be careful when considering objects, conditions, and phenomena.
Nothingness doesn't be, that's the definition of nothingness (at least in this question) that what is not, does not exist.
"Nothingness" is the condition of the absence of anything. In this sense, "nothing doesn't be" in the same sense that "running doesn't be" — it's not that nothing runs; its that running is an activity or condition for an object to be in, not an object in itself that may exist. Similarly, one may say that "sound isn't orange" — we're talking about a mismatch of categories.
If there was no Universe, if there was nothing, nothingness would be, that is the only thing that there would be.
Now you're talking about 'being' as a state of affairs which obtains; conflating an object's existence with obtaining of a condition of objects. They are similar concepts, which however apply to different things. An object may fail to exist (or rather, there may fail to exist any objects which satisfy some collection of properites); and a condition of such objects failing to exist may obtain. But conditions do not 'exist' except inasmuch as we use this turn of phrase to describe the fact that there is a situation in which the condition obtains.
This is, to be fair, a nuanced problem. Does fire exist? But fire is a phenomenon of rapid and exothermic oxidation — it is not an object but a condition which obtains of a system, e.g. a pile of wood or paper. Do apples exist? But apples are macroscopic conglomerations of atoms configured into complicated organic chemicals, including sugars which make the apple sweet, pigments which colour the skin of the apple, and fibres which help the mass retain a stable form — and the fact that these properties obtain is not a passive, but an active process involving electromagnetic forces on the nanoscopic scale. We abstract phenomena and conditions all the time and turn them into nouns, objects for our consideration and manipulation: rain, heat, winter, etc. are all nouns which describe conditions to which we are subject, rather than objects which we can e.g. pick up.
What of nothingess? This is different — it is a condition not of an object or the interaction of objects, but rather the condition that (in some region of space and time, say) there are no objects. If applied in the cosmological sense, it is the absence of any objects in the universe of discourse: the proposition that ¬∃x holds.
When you talk about nothingness "being", it seems to me that you are referring to the notion that this universe without objects still has facts: for instance, that ¬∃x. However, facts don't count as things which 'exist' in a first-order theory; and even if you move to second, third, etc. order theories, in which facts about one level are objects in their own right one level up, one can still have first-order nothingness (the absence of first-order objects) as a second-order object, without contradicting itself.