As a Briton, I would say that the answer is quite simple if the question is how a British person should spell the American expression “rain check”. One should spell it exactly as in the original, because one is quoting a word from another language. One cannot use it as an British English word because it has not entered common usage in Britain.
It is long British custom not to transpose US spelling when referring to US institutions, in contrast to the reciprocal American practice. Thus, whereas US newspapers refer to the “British Labour Party” as the “British Labor Party”, British newspapers refer to the “World Trade Center”, although they might write (incorrectly) that it was in the centre of Manhattan. A rain check is not an institution, but I would think the same principle holds.