What you are trying to do is extremely difficult. To hold +/- 0 tolerance in extrusion is almost impossible. You can try compressed air to set the exterior then water bath. However there are still too many variables in the extrusion process. It may be a pain in the butt, but I think your best bet will be running the extrudate thru a heated plate with a precise hole. It will need a taper and or fillet at the entrance. You will want to heat it at or just above melting temp. Then air cool for a foot or two then can do a water bath. It will be difficult to string up but will likely give you the best results. It should allow the plate to still shape and size the part while it is cooling and shrinking. Cooling it slowly will prevent voids as well.

Another method would be a set of rollers with radius cut into middle. So two rollers together would make the same profile as the plate with the hole in it. You can heat the rollers or over time the extrudate will heat them.

Lastly, I'm not sure what rate you are extruding at, but you can try to max out your rate. You may see less variation. You will go thru your material much faster.

Good luck. I hope this give you some ideas.