I think you are overthinking this one. Create a model in modelbuilder like the one below.
Source Data -> Make Feature Layer -> SourceData Layer -> Calculate field -> Source Data Layer
Set the expression for the Make Feature Layer to something like "DATEPROJECTCOMPLETED IS NULL AND PROJECTCOMPLETED1 = 1". So it only updates records where the date is null and the project has been completed.
Set the Calculate field expression to "time.strftime('%d/%m/%Y')" using python. This will add the current date to the records identified above.
Finally, export the script out of modelbuilder to python. Create a batch file which runs the script. Use the task scheduler to run the task once a day at around 11PM so it records the proper date.
Make sure the time and date settings are correct on the machine this is running on. May want to change the path environment variable to run Python more easily.