Decided to tackle the job this afternoon. Here is the old crown that I managed to pull off.
Here is the chimney minus the crown.
Here is the new crown mortared in place.
I also filled a couple small cracks in the mortar of the chimney, but it was in pretty good shape, so I didn't push it. Now I'll wait about two weeks and give it a few coats of a chimney water sealant. My perlite should be here in a few days, so I can get my liner insulated also.
The job is far from perfect, but it looks OK from the ground and will keep water out. What more do I really need?