How do I mirror/copy/save a harddrive from an old macbook that has an account in a logged-in state?
The real problem here is macOS isn’t designed to save run state to the hard drive in a persistent state. Hibernate mode doesn’t expect to be relocated on a different memory and does expect to power down completely and add RAM when the os is not suspended or hibernating. 

What you ask is trivially possible if you were running in a hypervisor such as VMWare fusion and could suspend and change the hardware from under the os due to a layer of abstraction. You’ll need an admin password to move forward, so that’s well documented. 

Make a new admin or follow the password reset procedure. Deal with lost keychain and make sure FileVault isn’t enabled before you decide to reset the password and move on is my advice. The OS isn’t designed to support what you contemplate.