Deffo poly in that application with that range of movement.
PU generally has a glass transition lower than your freezer will go so you wont crack it
It will get stiffer the colder it goes, but to be honest your best not putting too much heat into it while cutting as freezing really only works on small diameters and harder pu.
Freezing it to doof it in works a treat though.
There is a knack to machining soft polys, you need to get a cut on (don't be shy), and need to support to stop the deflection if its hanging out the chuck or vice
Use a tool with a fine honed cutting edge (no rough grinding), 2 ish mm tip radius with all the faces cut back to clear the poly rod and a narrow tip angle.
High surface speed and a light feed is a general starting point but hardness of the Poly will dictate this.
The poly should come away in a clean long ribbon and try to drag your face into the chuck
Milling is best as the material comes away in smaller chips/strips but if you dont have turntables etc then anything round becomes a ball ache...
If it comes off all scabby and stringy leaving a badgers arse behind you are doing something wrong
If you could lay your hands on a set of powerflex bushes to measure them up, you would be in a good position to try harder PU materials and cost it up properly.
Let us know how you go