Comments on this question considered whether the verb be could be modified by an adverb. This seems a question worth pursuing in its own right, so may I ask what completely modifies in the following sentence, if it doesn't modify be?
Whatever you choose to be, be completely.
Edit: And how about these?
He is almost a doctor.
Finally he is a doctor.
To be certainly can, because it has its existential meaning, as well as its copulative meaning.
To painfully be, not to painfully be.
I find it interesting though that while this suggests more motive for the dilemma than the original it weakens it not just in ruining the scansion (it's not like I thought I could improve on Shakespeare) but because the existential meaning is so much rarer than the copulative that the misinterpretation "to be or not to be what?" is easier to make here. (We could explore the effect of the decision as to whether or not to split the infinitive on various phrases that modify the existential to be).
Still, used existentially, be can clearly be modified, since it is a normal enough verb in function.
For a stab at modifying to be in it's copulative sense I would try:
I still am.
We gladly are.
In leaving out the predicate (having it deduced from a context that I don't give), I think it shows that what is modified is the link between the subject (I) and that missing predicate; that is to say, what is being modified is precisely what the copula represents.
This doesn't give a complete answer:
In the end, I'm left to conclude that whether we consider gladly as modifying are above comes down to why we are asking the question, we have different models for how language generally, and English specifically, works to solve different questions about it. It suits my needs (to have my hastily written pieces reasonably coherent and my more carefully drafted pieces coherent and convincing or evocative) to just answer "yes", but I won't claim to know how well this would fit with various models of grammar used by linguists.
Still, you can definitely modify the existential sense.
Allow me to answer your question with a question.
How could it not be? ;)
(Not here is modifying "be", in case it wasn't obvious)
Or for a different example, you can "be very", as in "Be very very quiet".
"Be" is a verb, like any other verb, and I see no reason why it could not be modified.
Granted, in your case, it is a rather awkward wording, because "completely" is an adverb that doesn't quite fit with "be". "Be it completely" seems to work much better, but I suspect that is because it adds a subject to the phrase.
As for your second two examples, there are a few options you have that avoid this conflict altogether.
"He is becoming a doctor". "Finally he became a doctor".
Yes, it is sometimes correct to modify "be" with an adverb.
Originally I wrote this whole post saying it is almost always more typical/preferred to modify the subject/object instead of the verb, as TecBrat mentioned in his/her comment. However, here is an example where it is most definitely correct to modify be, and not the subject/object:
Punishment should never be carried out in anger.
This completely modifies "be".
One should always be mindful to watch their step.
Seems equally correct as
One should be mindful to always watch their step.
SOMETHING, probably in some metaphorical sense of purposeful career- or character-building. What completely modifies is the verb phrase be (SOMETHING), where the parenthesized material is deleted by conjunction reduction, leaving be behind as a tag pro-verb, like the do action pro-verb in Do what I say, not what I do.