Sadly, because drives are encrypted to the console they're in, yes, buying another 2TB drive will be the quickest and easiest way to get up and running. If you have a large enough external backup, that won't be too bad, or as vulsker says, just backup your saves and then make your ISP work for its money by downloading games as and when you want to play them.
Only the target PS4 needs to be actually signed into PSN at the time. As long as the other PS4 is turned on, can connect to the same network and has the same account details, the target PS4 can find it and trigger the transfer.
If both are connected to the internet via WiFi, then for the transfer to occur, they also need to be hooked up directly to one another with an ethernet cable. The data transfer then occurs over ethernet, which can be quicker, with the WiFi simply the...
Sorry if you think that's misleading, it's just that Sony haven't called this one "PlayStation 4 Slim", it's just "PlayStation 4". It completely replaces the original model in the retail chain, so saying it's the "New PlayStation 4" is the most correct thing I could think of.
The Pro comes with 1TB built in, so unless you're selling the Batman PS4 before it arrives, you can just do a direct transfer and save yourself some time.