Having an interline agreement does not mean they will transfer the bag when the reservations are separate.

Unless something changed, the little train that goes between T1 and T2 has an interesting setup. One car is on the air side and one car is on the land side.

It seems China Airlines (CI) is the Ground Service Agent for PR in Taipei. So, if you see a CI employee, you can ask them I doubt it. Even if they said they could, I personally would feel more comfortable doing it myself.

It seems time will be a little tight. Focus on making the check-in deadline at the PR counter in T1. Good luck.