Success!!
I got a chance this morning to work on the rope. I tied a tight knot twenty feet beyond the two spots, secured the end of the core to a heavy piece of furniture, and pulled the cover back. The tow spots were easy to find, and they were caused by pitch that had soaked through the cover. A quick wash followed by a good rinse removed the pitch. I didn't see anything at all wrong with the core other than those two spots that are now cleared up.
The hard part was getting the cover all the way to the very end. With the aforementioned knot secured to the furniture, I worked the cover from the knot to the end. After a dozen trips, the cover still had an inch to go, and I had stopped making progress. I started working in small sections from the end back to the knot pulling the cover toward the end, and it took four more trips to get the cover all the way to the very end. I whipped the end with some 30-pound braided twine.
Since the new rope hadn't arrived yet, I reassembled the climbing rig, and enjoyed a nice 60-foot climb in an oak to check it out. I didn't notice any lumps or jerking on the way down, so I think the effort this morning was well spent.
Thanks again for the suggestion, Brocky. Now I've got a brand new rope AND a good backup.