Is there any way to do an in-place upgrade from Windows 7 Ultimate to Windows 7 Enterprise? I get an error saying that there's a compatibility issue, but perhaps there's some way to modify the registry to bypass this restriction?
You can use a registry edit to do it:
Change where it says Ultimate to Enterprise
Once you have done that load the Windows 7 Enterprise CD and do the upgrade install - Without UPDATES - and everything works just fine.
Once installed go to Disk Cleanup and do: Clean up system files - Previous Windows installations
No, this scenario is not supported by Microsoft.
Here is Microsoft's official list of supported upgrades:
note: this would technically be a downgrade rather than an upgrade