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?
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
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