Richard suggested that I convert my comment to an answer. Most of the key points are well addressed elsewhere, but I will repeat them here for the sake of completeness.
is there a good use case for using the C clef?
Absolutely. The problem is that keyboard culture has evolved to the point that the benefits of using C clefs are seen to be outweighed by the benefits of using only the treble and bass clefs and learning to read ledger lines.
Is it fair to encourage or challenge pianists to learn how to read in other clefs, or is it controversial?
It is both fair and uncontroversial. The problem is that it is also fair for pianists to refuse to accept the challenge, and that is what almost all of them will do. Further, it is uncontroversial because everyone more or less agrees that there's no reason for pianists to learn to play from notation that uses C clefs. As I noted on my comment, writing your piece with a C clef "will reduce the number of performances of your music, probably very significantly."
Has any composer written piano music with clefs other than treble or bass?
Absolutely, and this part of the question is the main reason for adding this answer. Most of Bach's harpsichord music uses the soprano clef for the upper staff. I'm less familiar with the music of his contemporaries, but I imagine that this was standard practice in northern Germany at the time. I don't know whether it was also standard practice elsewhere, or when it changed.
I do know that the use of C clefs for choral parts diminished beginning in the late 18th century, disappearing altogether probably some time in the late 19th century. This probably coincided with a rise in the prevalence of amateur musicianship, meaning that more people with less musical training were reading music, whether in the home or in choral societies. In Bach's day, a professional keyboard player would be expected to be able to read at least five clefs, namely the treble, soprano, alto, tenor, and bass. But professional keyboard players wouldn't generally be trained as keyboard players. They were trained as musicians, to sing and to play a variety of other instruments in addition to keyboards, especially strings.
The fact that keyboard music would use any clef in those days brings to mind another point: it's not particularly remarkable these days to see piano music with two bass clefs or two treble clefs.
When I first saw Elements in Space's answer suggesting putting part of the right hand part in the bass clef, I thought it seemed reasonable, and probably the best solution. But then I saw Matt Putnam's answer showing the right hand entirely in the treble clef, and it really does look entirely normal. Using the tenor F and E on the third ledger line below the treble staff is really completely unremarkable. Steve Bennett wrote in a comment that he is "pretty comfortable reading up to 3.5 ledger lines (eg, an A above middle C in the left hand)." This is true of most pianists.
In fact, when I was learning alto clef, doing score reading exercises years ago, I would often visualize the bottom three lines of the alto staff as the first three ledger lines below the treble staff. That is, the three ledger lines were the familiar thing that I related the alto clef to in order to familiarize myself with it.
It's a pain to write music like that by hand, drawing all those ledger lines. But who prepares performance materials by hand these days? Write your working score in alto clef if you like. Do whatever's easiest for you. You can even enter it that way into the computer to reduce the likelihood of errors, and then let the computer convert it to treble clef. But in general, when choosing clefs for a performance score, it's probably best to start by putting the piano part into a normal grand staff with treble and bass clefs, without octave transposition, and then adjusting only if it's difficult to read like that.
