Can I "hot copy" an entire running Macbook to a new one without using Target mode?
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.