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The list_of_elements_of_multiplicative_group should definitely return elements of R, not Python integers. Somewhat ugly, but ok there is your group:

sage: M = map(R, R.list_of_elements_of_multiplicative_group())
sage: M[2]*M[5] in M
sage: prod(M) == 1

Now the isomorphism with C2 x C4 is not unique, so its not clear how this can be improved. Maybe one could return a particular morphism from C2 x C4 to R. In any case, as far as I know this is not (yet) implemented.