Tim, you can check the rigidity of the compound/top slide to see if it is firmly supporting a cutoff/parting/grooving tool with overhang, particularly since the compound is located near the edge of the cross slide on a Smithy.
By placing a length of stock or a bar-shaped tool about 12 inches long in the toolholder (or turret), and applying light to moderate vertical finger pressure at the end of the bar, any movement that is seen in the compound would be an indication that the cutting tool has vertical movement onder the load of cutting. Besides the cutting tool being able to dip down while cutting, any sideways twist will very likely break a small cutting tool of that size.
If you're compound is set parallel to the spindle axis, you may want to try resetting it to be perpendicular to the spindle axis, or somewhere in-between.
I realize that supporting such a short/small part with the tailstock center on a Smithy is probably not possible. Attempting to use the extension for the tailstock ram will probably not be worthwhile, as the overhang is too great for the area of the base of the tailstock.
I agree with the suggestions of placing the cylinder on an expanding arbor, this would be a great improvement over trying to chuck a hollow part.
If you have a big endmill holder (and a drawbar) that fits the spindle taper MT4, it should be a good substitute for using a chuck. By turning a snug-fitting stem to put in the big endmill holder, a stub (the diameter that would hold your cylinder) could be turned for it to mount on. If you wanted to go further, you could drill the stub, tap for a 1/8" pipe plug or set screw, then slit the stub so it would expand inside your cylindrical part. Otherwise an end screw and a washer would probably suffice (with the motor running in the forward direction).