There appears to be no way to get an HDMI signal out of an iPhone at the same time as getting a USB signal into it. But there is one other way to get video out - using Airplay.

Airplay is Apple's wireless video transmission protocol which is uses to send video from iPhones, iPad and Macs to it's Apple TV box. One specific option - mirroring - will display the contents of the phone screen on the TV. I'm sure there's a delay, but it's imperceptible to me. So my new plan was to use Airplay to send the DJI GO's screen over Airplay to the Fat Shark goggles.

In an ideal world, the Airplay protocol would be published and well understood - but this is Apple we're talking about so obviously that's not true. And while people have certainly reverse engineered it over the years, every major update of iOS seems to break these 3rd party options. In fact, the only reliable way I could find to received Airplay was on an Apple TV (not even another iPhone could to it).

Now fortunately I did have an old Apple TV to play with since I had recently upgraded my older Gen 3 Apple TV to the latest. So I hooked that up to the goggles, mirrored my iPhone to it, and ... drum roll ... the DJI app was displayed on the goggles!