Your problem is a multi-budgeted matroid intersection with two budgets. There is a PTAS for that [1]. However, your case is so special, better algorithm exists.
You can assume there is only a single budget by truncating the rank of one of the other 2 matroids to $c_1+c_2$. Also, the weights are 0-1 weights. Observe the problem is equivalent to the following problem.
Given matroid $M_1$ and $M_2$ and $R, c$, find a common independent
set $B$ of $M_1$ and $M_2$, such that $|R\cap B|\leq c$ and $|B|$ is maximized.
The above problem can be solved in polynomial time. One can reduce it to a sequence of maximum weighted matroid intersection. Indeed, we give each element outside of $R$ a weight of $1$, and elements in $R$ has weight $\epsilon << 1$, and look for a maximum weight common independent set of $M_1$ and $M_2$. This will try to find a common independent set $B$ such that $|B|$ is maximized, and under that constraint, $R$ is minimized. However, it is possible that $|R\cap B|>c$, therefore we can truncate the matroid $M_1$ to a matroid with one fewer rank, and repeat the process. We truncate $M_1$ repeatedly until eventually we obtain $|R\cap B|\leq c$. $B$ would be the desired common independent set.
I have no idea what to do if you want maximum weight common independent set while requiring $|R\cap B|\leq C$.
[1] Chekuri, Chandra; Vondrák, Jan; Zenklusen, Rico, Multi-budgeted matchings and matroid intersection via dependent rounding, Randall, Dana (ed.), Proceedings of the 22nd annual ACM-SIAM symposium on discrete algorithms, SODA 2011, San Francisco, CA, USA, January 23–25, 2011. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); New York, NY: Association for Computing Machinery (ACM). 1080-1097 (2011). ZBL1377.90071.
