This should be easy to set up and script. First create the second layout. This could be just "duplicate layout" and give it a new name.
In the script, Have it do the following.
Duplicate record. This makes an exact copy of the record.
The use Go to Related Record and have it use the new layout and check "Show only related records" and "Match current record only"
This should take care getting the duplicate record on the new layout.
Ok, so that out of the way, I'm not sure that this is the best way of doing this. The duplicate record, may cause all sorts of new problems since it will be a duplicate, and if the original has not been deleted then you will have two records instead of 0. I think both results are probably not good.
Possibly you can set up a custom menu for the users that removes the delete record button and menu item. You could script a Cancel function that does delete an unwanted record. Once the sheet is created and accurate, just prevent them from deleting it.
Perhaps you can give us some more details as I think the problems you are facing can best be solved by carefully setting up separate Privilege Sets for the different aspects of the process.
How many different users are involved and what are the steps?
Who makes the request for Overtime? Is approval in advance needed? Who is recording the actual hours worked? Is it signed off for payment?
Controlling access to the different parts of each single record through customised privileges that prevent deletion would, as Bruce has already suggested, be a much better process than parking a duplicate record in another table.
I hope that helps,