Moving partial Site data between installations

I have a site that I manually migrated to a new instance using a simple mysqldump.

Long story short, poor timing with parties involved resulted in one weeks worth of site data not transferring over to the new installation, and staying on the old installation.
Now, the new installation has been collecting data, so I can’t just empty it and start over.

Does anyone know how I could import just that one week’s worth of missing historic data to the new installation?