The case of $R~$ bounded and $C~$ large is not covered by the formulas of Barvinok. The only published case as far as I know is for the row and column sums both being uniform. See E. Rodney Canfield and Brendan D. McKay, Asymptotic Enumeration of Dense 0-1 Matrices with Equal Row Sums and Equal Column Sums, Electron. J. Combin., 12 (2005) R29, Theorem 4.
http://www.combinatorics.org/Volume_12/Abstracts/v12i1r29.html .

To do the case of more general column sums asymptotically, you just need to apply a suitable central limit theorem. Each column corresponds to an $R$-dimensional random variable taking values in $\{0,1\}^R$. All these variables are independent, and the row sums are just the sums of these variables. You need a local limit theorem in the lattice case, for example around Corollary 22.3 of R. N. Bhattacharya and R. R. Rao, Normal Approximation and Asymptotic Expansions,
John Wiley & Sons (NY, 1976).

Don't expect any exact answers except for very small $n_i$. You can compute some exact values by making a recurrence based on adding one more column, and also by other means (see Section 6 of my paper).