I just wanted to say thanks to everyone that's contributed information to this thread (although some of it is misleading). I was able to connect my Neurio W2 to my Shed WiFi AP, while the Powerwall (actually the Gateway) is connected to a different AP with a different SSID. The APs are just connected with a switch. I did have to email Tesla support with the details, so they could manually adjust the setting in the Gateway, so it looked for the Neurio at the right IP address.
The biggest hurdle was getting a browser that could properly view the Neurio's website. Firefox failed to connect at all, and Chrome and Opera displayed the site poorly, and I was unable to load the Wi-Fi Settings page. I ended up using an old netbook, with an old version of Firefox, which correctly loaded and showed the available networks, and allowed me to enter the password for my Shed WiFi.
You do need static IPs (mine is done via DHCP reservation) for the Gateway and Neurio. They will both respond to pings, but the Neurio seems to only serve its website on its own AP, so it's normal that you won't be able to access the web interface once you have it connected to your own WiFi.
When you email Tesla support, they will need to know the IP addresses of the Gateway and Neurio, as well as the MAC address of the Neurio (it's displayed on the web interface, so take note of it before connecting the Neurio to your own WiFi). I also gave them my Tesla account email, which they can use to find/access your system.
I'm very impressed with the Tesla support. They responded within 3 hours, and did their magic in one go, along with the reply "I am beyond amazed at the level of achievement here, really you’ve done a great job, given installers struggle to do this."
I just thought I'd share my setup, now working, since a few people claimed it required the same AP/SSID and other things that didn't turn out to be true.
Neurio -> WiFi-2 -> Ethernet -> network switch <- Ethernet <- WiFi-1 <- Powerwall/Gateway