Thanks, that is clearer. I suspect that if you drill a 3mm hole then use a bit of 3mm rod it won't give much lateral support because of play between the rod and hole. Better is to centre drill and use a standard dead or live centre.
What sort of parting tool are you using – carbide? If so I can understand having problems with plastics because they need a very sharp tool. Best grind your own from an HSS blank – you can get parting blade holders for small lathes with narrow HSS blanks from the likes of RDG or Arc. Actually I see Arc have ready-sharpened blades.
Corian is a lovely material to machine and make small components from. Find a friendly kitchen fitter who will let you raid his waste skip.