Just make later pattern cartridges.
The type that uses two identical sheets of cut paper. I believe they’re called the 1861 cartridge in the ordnance manuals?
They are designed so you separate the pre greased mini from the paper and no paper actually ever goes down the barrel. These were designed to be faster to manufacture and were a lot less complex over older designs.
EDIT:
Here is how to makes them. You only need one pattern of paper, two pieces:
View attachment 75563
Nip the end of the paper with your teeth, pour powder down, and separate the mini from the paper. Stuff him thusly, cap, fire. It’s that easy. Also allows you to use a mini properly sized 1-2 thou less than bore diameter for top accuracy. No need to factor in paper thickness etc. as again no paper is stuffed down, just the bare naked greased ball. I have found unsalted lard thickened with some beeswax is a marvelous mini lube.