With the exception of the /home/oracle directory all other sub-directories within /home only accumulate to 1G in size therefore the oracle home directory is consuming the space.
Do I need to remount the latter dir as something else or is there an easier way of resolving this?
When accumulating all directories below /home, you have to count /home/oracle too, minus the size of /home/oracle/pcm/dbbkup, as this is as you stated a separate FS. The sum should be about 3,7 GB then.
You might have to do some clean up in form of moving, compressing/archiving or deleting files. If these are no option, you will have to add some space if possible.
If this is true then the problem should be visible in:
find /home/oracle -xdev -type f -size +1000000c -exec ls -lad {} \;
Note the use of -xdev to confine the search to the filesystem /home .
If you don't find any large files and have been deleting files, you may have deleted files which are open by an application in which case you will not get the disc space back until you close the application or reboot the server.
A quick way in Linux and most other SysV-based Unixes to find out which process is holding onto a file that would otherwise be deleted is the fuser command. it even has a -k option though I have always either manually sent signals or had a script to clear such out. Useful on old systems with many active serial connections that get clogged up.