Separating academic identity from social identity
Here are a few considerations:

Using a common name in place of a unique name will look unusual, since people who choose to publish under a variant of their name typically go in the other direction, to make it easier to identify their papers.  Everyone who learns about this will assume you either made an unwise decision about which variant to use, are trying to avoid prejudice based on your name, or are trying to disguise your identity.  In any case this change may attract attention, which is the opposite of what you want.

To separate your identities, you need to use your professional name in all professional contexts.  I.e., your university web page should use it, your colleagues should call you by this name, you should use it when you give talks or attend conferences, etc.  To have a successful academic career involving research you need to develop a scholarly reputation, and that can't happen unless everyone connects you with your publications.  That means either they connect the two identities or they know you under the name from your publications.  (In particular, it's not enough for you to be able to prove that you are the author.  In order for you to have a successful career, in the long run everyone must already know you are the author.  This is not such a big issue for graduate school applications, but it's a major factor in getting tenure.)

I wouldn't worry too much about proof of authorship.  This is not so easy to determine from the name anyway: even if you published your under full legal name, how would anyone know you weren't claiming papers that were written by someone else with the same name?  One big clue is your letters of recommendation.  If the letter writers discuss your papers, then everything will be fine.  (While if they don't discuss your papers, then that's a problem regardless of the authorship issue.)  At that point all you need to do is include a sentence in your publication list saying "My publications are under the name X."  You don't need to explain why, just to clarify the facts so nobody is confused.

Once you have started publishing, substantially changing the name you publish under is highly disruptive.  It can be done if necessary, but it may hurt your career by fragmenting your reputation, and the only way to combat this is by loudly and publicly connecting the two names.  This means you should try your best now to choose a variant of your name that you would be happy using for the rest of your career.

It's worth thinking about whether you need to do this.  In my experience, academia is tolerant of eccentricity and impressed by confidence.  If you treat this situation like it's no big deal, then I'd guess that most people will ignore it and anyone who tries to gossip about it will risk looking provincial or small-minded.  However, this of course could depend on the details of your situation.  (And there's also the issue of students: first-year university students are on average quite a bit less mature than faculty.)