Clamp the tube right down on the table, right over the center T-slot. Sometimes I use a piece of shafting through the tube as part of the clamping system.
Thanks everyone, I really appreciate all the suggestions. I'm thinking Jim's suggestion is probably the most viable for my setup...
Although I probably should have thought about using a V-block, I can see where three clamping edges (and the aluminum insert) could have provided enough force to keep the paper from tearing.
I finished this tube prior to asking my question, but I ended up just spotting and drilling out the corners of the sections I had to remove, then cutting between the drill spots and filing to fit. That worked fine for what I had to do this time, but I'll eventually need to mill out straight, thin slots as well.
Here's a pic of what was dealing with:
As you can see, that slot ended up wandering quite a bit (and I wasn't getting a very clean cut at all. Switching to just drilling out the corners and connecting the dots with a cutoff wheel ended up being much easier (but won't really work well when I need to cut a thin slot).
For the slot in the pic above, this turned out to be the culprit: