I used to brush my necks out too, but stopped a while ago and didn't notice a measurable difference in consistency. I reckon the collet die does a pretty good job of crushing any carbon in the neck into "seating dust".
For your 6br collet die, where there's a will, there's a way. Without having a first hand look and a bit of a measure myself, I won't tell you that removing material off the bottom of the die body and collet will work, but that's what my plan A would be if it was me and it looked like the 6br will physically fit into the bare 243 collet up to the neck. You will also need to shorten the mandrel, whether that is by cutting off the bottom of the mandrel and losing the decap pin flash hole centering, or the top of the mandrel and peening a new step onto it is also another thing to think about.