It's starting to sound a bit like a big CD.
PTFE Filled PC could possibly do the trick.
I would diaphram gate into the spiggot internal surface, with the material flow aimed at the cylindrical wall of the spiggot rather than at the opened cavity of the disc.
I would keep the wall section of the cylindrical portion equal to the wall section of the disc, with generous and concentric radii at the transition.
I would not fill it to fast, as that would require excessive pressure, possibly ressulting in overpack near the gate, but not as much at the extremities.
I would use a hot mould, medium melt temp, medium back pressure (so as to get a homegenious melt) and a slow to start injection speed, ramping up to about 80% full, then slowing down so as to prevent overfilling and blowing the mould opened. As the speed dropped off, I would still keep the pressure up, so that the mould packed in a controlled but firm manner. It is essential to maintain consistent and uniform pack, and consistent and uniform mould surface temperature.
It will be necessary to take care in the thermodynamics of the mould at design stage.
The moulds and techniques used for the manufacture of CD's should give a good starting point
Regards
pat