This is not an answer but a collection of comments based on my personal feeling, but I post it as an answer because it is too long for a comment.
First, here are two clear facts:
- のある simply does not have the same meaning as である. ピアニストのある私の姉 is incorrect.
- Replacing AであるB with AのB sometimes causes ambiguity. For example, ピアニストの姉 can mean either “(my) sister, who is a pianist” or “a sister of a pianist” depending on the context.
Now I will move on to a less clear part.
I agree with you that AであるB sounds a little too formal for a casual conversation (although I would not call it “literary”). In a casual conversation, I would probably avoid a relative clause in this case and say something along the following.
My sister is a pianist, and she is always careful not to injure her hands.
The same meaning as AであるB can be also expressed by AのB in some cases, and I agree that AのB can be less formal than AであるB. But in your case, I find using の in place of である less natural:
Unfortunately, I do not know why I feel it less natural than ピアニストである私の姉.
ピアニストであるseems ambiguous. That is, it seems like it may be applied to either
私の姉. Of course the context makes it clear, but in another situation, would there be a better way to specifically indicate the sister? Maybe
私のピアニストである姉? Although that sounds funny/wrong to me.
私の、ピアニストである姉は(of course in this particular case 私の itself may well be unnecessary). However, I wouldn't say that it's better, unless confusion is a serious possibility.
NのNis acceptable in place of
NであるN. (I am not sure what counterexamples exist, if any, so I can't write an answer at the moment.)