Can I "hot copy" an entire running Macbook to a new one without using Target mode? Is it possible to copy a running Macbook's state (running Sierra) over to a new Macbook?

I can't use target disk mode for two reasons: one, I don't have a Firewire cable and am not sure where to get one on my tiny island;  two, the old Macbook is broken: when restarted, it usually won't boot, and I fear it won't boot even into target disk mode. (At the moment, I'm keeping it running until the replacement arrives. Day five so far.)

There is a Time Machine backup, but I'm excluding a lot of non-essential files from it. If I added all those files just to transfer everything over, it'd enlarge the overall backup size forever, which I'd like to avoid.
You can't clone a fully booted Mac to another completely booted Mac


Not completely accurate.

A fully booted mac can be copied, in it's entirely, to somewhere else. Use Carbon Copy Cloner (easy) or rsync (Unix guru).

The only place "somewhere else" cannot be is a booted volume. The hardware hosting the destination can be booted to the desktop, you just need a second partition on the drive and send the clone there.

You can also clone to an external drive or a disk image. In your case I would clone the old machine to the external drive that has Time Machine on it (NOT as a disk image), boot the new machine from the external, and clone again. Use Carbon Copy and it will not bother copying files that are already there.

You can do the clone-to-external part right now.