What would prevent me from copying pads (or other components) in Altium?
Shouldn't be a problem if all components are "not locked". There also needs to be the same stackup (layers) between projects. I copy parts routinely between projects and I've never had a problem. 

Try copying the components that will not copy one at a time (use the relative reference of copying to make sure they end up in the same place).

The easiest thing might be to copy the PCB file from one project to another and just use that to start with.