FYI, I've used Inventor for close to 20 years now, and I've never seen anyone use that "Owner" property for anything.
What I did was write a relatively simple iLogic rule that looks at the file name of the active Project File and copies that to the existing "Project" iProperty in the drawing that you mentioned. In my case, I set an Event Trigger for this rule as "New Document". This means the rule would only trigger when the document is first saved. I had reasons for doing that in my application, but you may want to use "Before Save" instead, so it will update every time the drawing is saved if the project file name changes.
Dim IPJ as String
Dim IPJ_Name As String
Dim IPJ_Path As String
Dim FNamePos As Long
'set a reference to the FileLocations object.
IPJ = ThisApplication.FileLocations.FileLocationsFile
'get the location of the last backslash seperator
FNamePos = InStrRev(IPJ, "\", -1)
'get the project file name with the file extension
IPJ_Name = Right(IPJ, Len(IPJ) - FNamePos)
'get the project name (without extension)
IPJ_ShortName = Left(IPJ_Name, Len(IPJ_Name) - 4)
'get the path of the folder containing the project file
IPJ_Folder_Location = Left(IPJ, Len(IPJ) - Len(IPJ_Name))
'Converts the project file name to UPPER CASE
IPJ_Shortname = UCase(IPJ_Shortname)
iProperties.Value("Project", "Project") = IPJ_Shortname
iLogicVb.UpdateWhenDone = True
I'm not sure why lines 13 and 14 are there. They are determining the folder location where the project file is stored, but then nothing is done with that information as far as I can tell. It's possible that they are leftovers of an older version of the rule. They could probably just be deleted, but I didn't have time to test that and troubleshoot it before posting this example for you, so I left it as-is.