Apple - 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.

You can't clone a fully booted Mac to another completely booted Mac. At least some files and folders won't be overwritten because they are in use (e.g. the Mach kernel file).

Without some direct connection (e.g. Thunderbolt cable, Firewire-to-Thunderbolt cable, Firewire-to-USB 3C etc.) you are left with either the Migration Assistant or a Time Machine backup.

Your best bet is to make a complete Time Machine backup, boot the new Mac in Recovery Mode, attach the backup drive and restore the content of your old Mac.

Using the Migration Assistant and a network connection is probably rather slow because it will rely on Wi-Fi (assuming that the Wi-Fi chip in the old Mac is not as performant as the one in the new Mac).