When I teach Alg Top I
(This is a corrected version of my original, I do the Kunneth and UC Theorems for rings off-the--cuff answer).
If R of hom has proj dim 1 and C is a flat chain complex, since that's all then you use to get the short exact UCT sequence. I seem To get the splitting, you need to recall also assume that C is projective, so that the unnatural splitting requires R map C -->> B from the complex to be the subcomplex of boundaries has a PIDsplitting. (Sorry I don't have time to check this right now.) But this suggests how unnatural
A spectrum level construction of the splitting really is (assuming this is correct)seems unlikely since you don't have kernels and cokernels, but only fibers and cofibers there.