Compatible seatpost for Brooks C13 saddle with 7x9 oval carbon rails?

Carbon saddle rails are made oval to increase strength and reliability, and you can get these types of adapters for some seat posts. I am not sure how cross-compatible those adapters are, but it may be an alternative to spending more money on a new post. I do recommend using a torque wrench any time you apply force to a carbon component just to avoid over-torque. I looked at the spec sheet on Ritchey comp seat posts, I didnt see anything that led me to think it would work any better than the post you already have, but I have always been a big fan of Ritchey's products, and they have great customer support so it wouldn't hurt to give them a call or shoot an email and ask. You might consider a Thomson Elite post as I think you can buy the post with whichever clamp best suits your needs and the clamps are interchangeable which I find nifty. I love these posts, I have three of them and they never die or malfunction, and they are almost as light as carbon, easier on the wallet too.

Saddle rails converged on a 7mm diameter round rail and stayed there for quite some time. You can make them out of solid steel, or tubular steel, or solid or tubular titanium. Then we came to the carbon fiber era. I don't have solid documentation of this, but I'd guess that 7mm round carbon rails can't be made durable enough.

So, we came to the 7x9mm standard. I don't have my hands on such a saddle. However, it seems like (as stated on forum threads like this one, this one, or this one) these rails are 9mm tall, but 7mm wide. The tops and bottoms of the rails are circular, and many seatpost clamps should fit them.

Now, why many and not all? Many seatpost clamps cover the rails on top and on the bottom. Some clamps enclose the rail entirely. For example, Enve's clamp is shown below:

Thus, they ship two separate clamps for 7mm round rails and 7x9mm rails with each post. Alternatively, Kent Eriksen's titanium seatpost doesn't fit oval rails at all, and they say so on the product page. Basically, any seatpost where the bolts clamp side to side are not likely to work unless specifically designed (e.g. Enve). Or, sometimes a saddle's bolts won't offer long enough threads to engage fully if you put a carbon-railed saddle in there. I'm not able to find specific documentation for this, but I believe Thomson seatposts may sometimes be in this situation. Thomson offer a 7x9 (or 7x10) clamp on their site, but I have heard people say their stock clamps work fine. In any case, I believe the Ritchey clamp should work, as it clamps from above and below the rails.

So, why does the OP's original seatpost not work? That's a mystery, as it seems like it should from the description. Some photographs might be helpful if the OP wants to diagnose the issue.