Is there a higher Belyi theorem?

I was thinking the other day about this old question, a favorite of mine. Here’s another version of it — if anything along the lines of Belyi’s proof of Belyi’s theorem works, one might expect this to be true. Question: Suppose given five points p_1, .. p_5 in P^1(Q(t)). Is there a map F from P^1 to P^1 such that the the critical values of F and F(p_1), … F(p_5) are all contained in a set of size four? This would be “Belyi for M_{0,5},” I suppose. I have to admit, I feel like the answer is probably no. Why it seems hard: every set of p_1, .. p_5 obtainable is obtained from some family of covers of P^1 branched at four points — in other words, it’s the image in M_{0,5} of some 1-dimensional Hurwitz space. So you have some countable union of curves in M_{0,5}. Now all you have to do is write down a curve in M_{0,5} which is not one of these. But… how do you know you can do this? If you were allowed to work over C, it’s easy! But there are only countably many curves in M_{0,5} defined…