Reporting back:
I swapped the Pi2B for a Pi3B, plugged everything into the Pi3B except the WiFi dongle as the Pi3B includes WiFi. I moved the SD card across, and booted up.
Bingo, 360 Radar display. Configured the display how I wanted it, then shut down Chrome so it memorised the set up.
On next boot, everything is happy. I have 360 Radar and the METAR screen on a loop, 45 seconds of each.
I'm using Raspian Buster standard version.