It would help greatly if we knew what type of 'plastic' the material is. Since you are calling it 'tube' and have sort of confirmed its soft material (by positively acknowledging Howard's stiffening thoughts) then possible contenders for 6mm (OD or ID?) tube might be Nylon, PTFE, PFA, PVC or something harder like Polycarbonate.
I presume that whatever you are making needs to have a large length to diameter ratio and that the tube has to pass through the headstock spindle? I can think if no other reason for not using solid material and drilling a hole through.
Am I correct in assuming that you are wanting to make a large quantity of whatever the item? Why otherwise would you need to a number of tests to find the optimum cutter?
Obviously the replies, or some of them may have helped you but by providing more detail with your initial question the help offered will be much more focussed.
Ian P