Part of the problem with a 3 jaw is also the squirm effect: when one of the jaws opposes the tool, the workpiece is stiffer in that plane, than when the work rotates another 60°, where nothing is opposing the tool. So you've got a cyclic 3 lobed vibration induced in the workpiece every revolution. Combine this with the end thrust of the tool, and you create a definite squirm effect. If the jaws are smooth, it takes enormous squeeze to positively stop the piece from moving axially.
Most likely, you will damage the scroll if the chuck is small for the task and you try to really crank it up. Better to use an endstop, or a roughing jaw of some sort.
On our Summit lathes, with 4.125" spindle bore, we've always had this slippage problem, because we don't want to change 12" chucks back and forth on a whim. But because we can put large diameter heavy bars through the spindle, we mounted another 12" chuck on the outboard end of the spindles. If the work is long enough to be held in both chucks, it doesn't slip through, but for shorter pieces, we mount a stop bar held in the rear chuck, to butt the workpiece against, while only the front chuck does all the driving. That also eliminates the work pushing through.
Edit: if you know anybody who has a Rocklinizer, you might get them to rocklinize the jaw grip surfaces. This amounts to a very fine grit roughing up of the jaws that may last a while. However, the carbide particles that get embedded in the process will raise likely raise hell on polished surfaces such as chromed rods.