When is it acceptable to issue an update to a manuscript that is still under review?
I think you are focusing too much on the details, which may be confusing the issue.  It sounds to me like your goal is to make sure reviewers of paper B are convinced by your arguments, rather than to write up a proper proof of A.2.  

It is a perfectly acceptable and common thing to refer to paper A for a proof of say Lemma 1 even if Lemma 1 is not explicitly stated or proved in paper A.  (E.g., "by a mild extension of the proof of X in [A], one gets...")  What is important here is two things:


your explanation of exactly how it follows from [A] is clear to someone who has not read [A]
that there really is nothing new to check and you are not just being lazy


From what you wrote, it sounds like at least the first part was an issue for you.  And if you can't make the proof of your lemma transparent in such a way, then you should write down the proof, either in paper A or paper B (possibly in an appendix).  

Since you've made it clear you don't want to do the latter, and this sounds like a minor change to paper A (just adding in a small amount of details), it's fine to revise your paper A and update it on the arXiv (so make sure your paper B references the new arXiv version) but there's no need to revise your submission of paper A for a minor change.  You can just point out you made this change upon resubmission after getting the referee reports.