I have problem with "ORACLE-00602: Cannot find ORACLE_HOME for SID " " in the Windows Registry" with SMO 3.0.3 and 3.1 but I am not sure same as your, environment as below.
Windows 2008R2 EE SP1
Oracle 11gR2 X64
I found below link describe Oracle 64bit registry in "HKLM\SOFTWARE\Wow6432Node\ORACLE\ORACLE_HOME", but 32bit application cannot recognise. I cannot found SMO x64 access Oracle registry information and I try export registry HKLM\SOFTWARE\Wow6432Node\ORACLE\ORACLE_HOME and edit location to HKLM\SOFTWARE\ORACLE\ORACLE_HOME, after import new edition and the problem disappear.
I ran into this issue as well after upgrading Oracle 22.214.171.124 to 126.96.36.199. No resolutions found on the net helped.
error [ERROR] SMO-05075: Profile create failed: ORACLE-00602: Cannot find ORACLE_HOME for SID "DELTEKCP" in the Windows Registry.
Setting ORACLE_HOME wouldn't resolve it. I had a cloned box that didn't have the upgrade yet.
I exported the registry from that box for the key [HKEY_LOCAL_MACHINE\SOFTWARE\ORACLE\KEY_OraDb11g_home1] and imported that into my upgraded box and this RESOLVED this issue. I'm not sure which key resolved it. I'm attaching a screenshot of the keys in this export for reference.