So, I did something stupid.
During the course of migrating a Windows Server 2008 domain to Windows Server 2012, I rolled back a snapshot of a running DHCP server to a fresh install state after having authorized the server in AD. This effectively destroyed the server as AD understood it.
Then I rebuilt the server using the same name and tried to authorize it again. AD wouldn’t have it– according to it, a server with the same name was already authorized for DHCP.
The specified servers are already present in the Directory Service
Facing a very long-winded fix involving ADSIEdit and waiting for AD replication cycles to complete, another engineer pointed out in the comments that all you have to do is restart the DHCP service (not the entire server) on the box that’s failing to authorize and try authorizing again.
In my case, it worked!