I still don't think, that the resulting DVR is then in $k(C)$ or am I mistaken? Sorry but I'm not very familiar with x-adic DVRs so I need some time to understand your example anyway.

Well, I had forgotten to mention this explicitly. But still thanks for your answer. It is still interesting, that (at least) the assumptions I made are necessary.

First of all I want to thank you for the answer. I'm not sure if I understood your example but maybe I haven't made clear, that I want the DVR containing $\mathcal{O}_{C',Q}$ be itself contained in $k(C)$. Is this satisfied in your example? (I'm not sure but I think it is not)