In Native Son Bigger's principal fear, at least on the surface, appears to be that of losing face in front of others in the neighborhood. In the pool room, he provokes a fight, ostensibly because he believes the others are too cowardly to carry out the planned robbery of Blum's store, when Bigger himself doesn't want to go through with it. He then cuts the baize top of the pool table, causing the poolroom owner Doc to pull a gun and throw him out, because this is a presumably safer way of showing a kind of street bravery.

In my opinion, however, Bigger's greater underlying fear, and the one that drives all of his actions, is that he will never be able to escape from the neighborhood and the empty, violent life he leads. He dreams about flying a plane and revels in the detective stories he reads in magazines. He inwardly expresses a kind of admiration for the fascist dictators in Europe because they have the ability to whip their people up into a frenzy of action. Bigger has a despairing fear that the purposeless, poverty-stricken life of his family and others on the South Side is a permanent thing, that there is no remedy for it.

When Jan and Mary behave (without knowing they are doing so) in a condescending, insensitive way to him, Bigger's feelings of fear and anger are brought to a head. Though he does not intend to kill Mary, after he's done so he feels liberated in some sense, having created a new life for himself, though he knows that eventually he will be arrested and executed. Bigger's fears are thus confirmed; he is unable to escape the life he's been sentenced to by a racist society.

I think that there are many reasons that could be used to explain his main fear, as the subtext of the question indicates.  I think that White society does seem to be Bigger's predominant fear.  There is a fear of what White society thinks of him when he cannot say anything to Mary's mother about him being in the room.  His fleeing from White society's police and the relationship and perception he has about White society all result in his fears of it.  His fears are certainly justified and well grounded in what is the reality for someone in Bigger's condition.  Yet, they are fears and Wright might be using this to explore how African- Americans, particularly African- American males, perceive White society.  The odd element in this configuration is that Bigger does believe in the American Dream.  The belief that Bigger can actually find his own niche in the American Dream is one that inspires him to get the job that enables him to be near Mary in the first place.  The fear of White society is enhanced by his desire to be close or to be a part of it.  In this light, one can see his fear as a twisted condition of reality within it.