*Calling* support seems to have fixed it
I would never do this, but actually had to call the support phone and laboriously typing in the activation code. After that I was transferred to a support person. Since my local one is closed it was transferred to a person with a very pronounced Indian accent. They were nice and just told me to wait while they verified the code, then asked me to re-enter the license again, and this time it actually worked.
So it might help to go the old-school route and just call support.