We have multiple P/N’s that all fall under the same Project. Is there a way to do this in CTS or does each individual P/N require a different project and therefore a different FMEA/CP?

Unfortunately no, each individual Part would require a project and different form. 

You can probably try to define a product family and create some records in reusable library for this product family. It should help a little.

This way you won’t need to add data each time and can just select rows from reusable library for these projects.