As I see it, you can't press the finial tube in from the center of the tube out to the end because you would ream the entire tube with the thickness of the finial pressure assembly. Then your center ring assembly wouldn't fit when you attempt to press it in.
What you might try is predriling a small recess inside the tube to hold the threaded end of your finial tube. You would need to grind the end of the final so that it would freely fit inside the 10mm tube and then attempt to epoxy the threaded portion of the finial tube into the pre-drilled recess. This would lead to a concern over the longevity and durability of the epoxy finial tube. Over time, would it stand up to the normal wear and tear of pen usage?
Have you considered turning an entire new finial cap out of the same material as the pen? I know the break in the pen's exterior is what you are wanting to overcome and avoid, but the cigar pen may not be the best kit to utilize for a closed end cap. You may have to design an alternative transmission engagement mechanism.
Now, with those caveats said, you can accomplish what you want by cutting off the end of the final tube from the threaded and pressure fit portion of the finial tube assembly. Then drill your blank about a 1/4 to 1/3 to accommodate the 10mm tube (or to a depth sufficient to accommodate your pin chuck). Then drill out another hole down to the final depth with a bit corresponding to the final tube outside diameter. Don't know that off the top of my head, but I'm think it is either a 7mm or 'O'. Pull out a micrometer and check it (I'll do that tonight and check). Then epoxy the 10mm tube in. I would wait and glue the finial tube in after turning with pen chuck (voice of experience there). The challenge is getting the depth correct. Too shallow and the pin chuck won't work. To deep and you have to material to glue the finial tube into. Remember that the finial tube is now shorter than originally designed, so take the finial cutoff into account. If you have another tube with the same interior diameter as the finial tube, then this is a cakewalk and you only have to worry about the finial tube being too long.
After turning, you'll need to glue the finial tube in and get the correct depth. Not too tricky, just test and set everything before gluing. A waxed wooden dowel helps with the insertion.
I would practice this a few times on my scrap blanks before attempt with my final blank.
post pics when you get it right