Unable to re-install an instance using the original instance name
The cleanest solution IMHO will be to remove reporting services, too, as part of cleaning the old instance off the machine. Until you've cleaned off everything service-related, that instance name still "exists" - and it can be a really thorny and frustrating experience trying to remove all the registry entries, GUID by agonizing GUID.

I suspect that SQL Server must keep some things around e.g. for connectivity to SSRS from external sources via the instance name. This is why you can't just pave a new instance over top of it.

The other alternative would be to leave your SSRS instance intact, but install the new instance with a different name, and just update application connection strings, aliases, DNS etc. so that they will connect to the new instance name (hopefully this change will be as transparent as possible).