"Bewaring" is an antiquated and basically obsolete form of "beware", which is almost exclusively used as an imperative verb ("Beware!") or command ("You must beware the tides of war!"), these days. That is how "beware" is defective, here: the inflected -ing is never used. I, certainly, have never seen it used in modern prose, nor heard it used in modern speech.
If one wants a progressive form of this verb, then "being wary" would be what one would say. Similarly, "You must be wary" is an exact synonym for "you must beware". "We were being wary" would be the past progressive of the verb "beware"; but one won't really hear "we are being wary" or "I am being wary", because to "be wary" tends to mean to be completely silent and focused on one's objective (which is often to remain unseen/unheard/out of danger), or to stay focused on some difficult, physical task at hand (which makes it inconvenient to speak). So one almost never hears the first person, present progressive form of "to beware". This is almost certainly the reason why the verb is "defective", as well.
I think the main reason people don't use inflected forms of this verb is because "wary" has largely dropped out of use; these days, people will say "aware", which is the original word ("ware") with an added prefix ("a-", meaning here "to be in a state of", like the colloquial "a-running", "a-changing", etc). Typically, the present progressive inflected form of "beware" will be "staying aware", or "keeping aware", or even "staying awake", or "keeping on top of things", as in:
While keeping aware of the traffic before you, look in your rear-view mirror and check to see if the cycle lane is clear, and only then make your right turn.