liverwort,
I wouldn't be as concerned about using the same cordite as I would the primers. I am not concerned about loading Berdan primers "per se" but the 303 British ammo may be loaded with varying diameter primers and may contain mercury in the priming compound. Primers being corrosive can be dealt with pretty handily, but mercuric primers destroy the cases for further use by embrittling them. I load several different calibers with Berdan primers but depend on an old stock of common diameters primers that I have been to find replacement for. My online searches for a replacement source have been unsuccessful, so far. If you have the correct diameter non-mercuric (and preferably non-corrosive) primers, you can reload the cartridges that didn't fire and any that you haven't tried, if you remove the original primers by safe means. The cases from the cartridges that did fire may or may not be useable. This sounds like something I would do but I believe using reasonably available modern cases would be much easier. It is my understanding that when cordite was used the cases were charged before "necking" the case down to the final diameter. As you have likely discovered, shaking the cordite strands out of the cases would be hard enough but putting them back in will require a lot of time and must be done "by hand". Unless you can salvage a lot of cordite, I believe it would be more economical to salvage the bullets by a collet type "puller" and forget about reusing the cordite. This would be true if you can find the correct Berdan primers or you use Boxer primed modern cases. I think the wad was used to "locate" the cordite strands during "necking" operation, rather than a filler. Any cases you would use would already be "necked" down. If you think a filler is required, that would be a decision no one but you should make.
Mike