Many apps will just use the standard camera api. Maybe contact some camera app developers to find out. That way your results will be the same as using the built-in app.
Alternatively if you know someone who's able to write iOS apps you could get a simple app built to trigger the built in app is via the API.
Another idea would be to mount the phone in a tripod mountable case and then to weight the tripod to such a degree that a light tap to activate the shutter wouldn't move the phone
Thanks for your great suggestions! I'm not convinced enough that the third-party apps are using the same API that the native app uses, and I cannot afford for there to be any discrepancies. In addition, I need to test the native HDR Photography feature within the native app, which hinders any possible workaround using the third-party apps.
Your suggestion to program a bespoke application to trigger the shutter is a good one. I have a computer science background, so should be able to put something together for this.
The easiest solution seems to be your third suggestion: stabilising the tripod as much as possible, and minimising the force applied to the screen-based shutter release. It would have just been nice to completely remove the possibility of camera movement by way of some remote shutter.
I can see why the AppStore cannot provide a solution to this, since these developed apps can only control their own functionality and cannot modify Apple's Apps or functionality. I can therefore only hope that Apple release support for this in future software updates, perhaps allowing a setting change to use the headphone play/pause button to act as a shutter release. There is indeed an app for Jailbroken iPhones called 'CameraButtons' that allows the assignment of the shutter release to the volume up/down buttons on the handset. If only Apple would release a software equivalent...