Is the phone detected in normal mode or DFU/recovery mode by iTunes when the copy screens are connected?
If you are doing data recovery, I would forget about the copy screens since you have image on the original screens. Troubleshoot the touch lines on the logic board. To get you started, the power for the touch system comes from a chip called U3 on the 5c---it is the display/touch power ic. Look for lines that are SAGE (touch) 5v7 and -5v7. Troubleshoot from there.
good luck---your question was very clear, I am happy to help.
Jessa