[...] suggests to me that what I wrote is indeed sufficient due to the separability.
This is not true. Indeed, by (say) Theorem 2.3.1 in Bogachev's book, the operator $R$ must be nuclear.
If this is false, then I am happy with answers saying it is true but you need these extra requirements on $\mathcal{B}$ or $R$.
As noted in Remark 3.11.24, with further references there, "the class of all nuclear, symmetric and nonnegative operators between $X'$ and $X$ coincides with the class of the covariance operators of Gaussian measures on $X$ precisely when $X$ is a space of type $2$."
As stated on p. 213 of Vakhania's book, "the [general] problem of d[e]scription of Gaussian measures [in terms of the characteristic functionals] is equivalent to the problem of description of the class of covariance operators of Gaussian measures. In the case of general Banach spaces the above important and interesting problem is still unsolved."
It appears to be unsolved even now.