Ahh, that's helpful. I see now that when I "click and cancel", the Label Merge option remains selected.

So now I have an accessible Update Labels button, and when I click it the first page changes from:
Name1 - Name1
to:
Name1 - Name2
(which is correct)

But then when I scroll through the records I find the second page shows:
Name2 - Name3
instead of:
Name3 - Name4

Which is still the basic problem I had with my OP.