When I teach Alg Top I, I do the Kunneth and UC Theorems for rings R of hom dim 1, since that's all you use to get the short exact sequence. I seem to recall that the unnatural splitting requires R to be a PID. (Sorry I don't have time to check this right now.) But this suggests how unnatural the splitting really is (assuming this is correct).

(This is a corrected version of my original, off-the--cuff answer).

If R has proj dim 1 and C is a flat chain complex, then you get the UCT sequence. To get the splitting, you need to also assume that C is projective, so that the map C -->> B from the complex to the subcomplex of boundaries has a splitting.

A spectrum level construction of the splitting seems unlikely since you don't have kernels and cokernels, but only fibers and cofibers there.