Is it possible to remotely take a photo on the DJI Osmo Mobile (iPhone 6S or 7+) - possibly by using an second iOS device?

This would be while using the Extension Stick, as an example.

I think it is OK since a bluetooth host (iPhone) can connect to many bluetooth deveices. For example, you can connect your iPhone to your Osmo Mobile and meanwhile an extention stick or a headset and use them to control the shutter of iPhone.

I think some 3rd party app might allow you use one iPhone to control another iPhone's camera.    I'm sure it can work on Android phones with a rooted system.

You can't use DJI Go to do it though, nor I see much point in using DJI Go for shooting photos only.