I have successfully threaded acrylic acetate (AA), Alumilite, and Rhinoplastic without problems on a metal-working lathe turning headstock by hand.
Rhino is the most challenging. . Alumilite is easiest. . Rhinoplastics can vary: different colors ... different brittleness (don't know why).
Have not tried threading the (more) brittle Inlace Acrylester from PSI.
If possible, choose a thread pitch that is coarser that you would use on metal materials.
Take it slowly and do the threading in small steps, using adequate lubricant. . I use (lots of) WD-40.
Back off frequently to remove waste, but avoid complete removal of the tap/die (restarts never occur perfectly). . Allow for imperfect end threads.
Take extra care with centering at the time of initiating the threading. . Do the threading by hand on the lathe (not powered) to help with this.
Avoid forcing and allow the tap or die freedom to "pull itself into the workpiece".
Use a die-holder on the lathe. . Little Machine Shop sells one; I use one of these. . Rick Herrell also does, I believe (but not sure).
Mount the tap in a Jacobs chuck which is initally loosely mounted in your lathe tailstock ... loosely to allow the tap to get pulled into the work.
As the threading progresses, adjust the tailstock wheel to just take up the slack that occurs as threads are cut.
The tailstock should not be tightened down, so that it can easily slide when backing off.
When tapping, try to avoid the need for a bottom tap and design your work accordingly.
Do everything gently/cautiously, turning the lathe headstock by hand ... NO POWER to the lathe. . Lightly grip the Jacobs chuck when tapping in order to prevent it from turning, but allowing it to pull into the work, or to back off.