Have both use the same specialised libraries.
If anyone wants inspiration, Winamp is the gold standard for media players with excellent GUI bling and skinning.
If your test involves two different GUI systems running the same 'specialised libraries', you are running the specialised libraries, not the GUI systems.
A media player involves heavy crunching and streaming and memory I/o, and access to the GPU if possible - these things that game engines excel at.
Most GOOD game engines are in fact an entire operating system, abstracting away the host operating system, and attempting to depend on it very minimally - giving coders an environment which does not have to conform to "business user" semantics.
This doesn't mean game engines don't burn through energy. But on mobile, at least, most of the good game engines do allow you to fine-tune your frame-rate such that, indeed, you don't need to update anything - video or stream - at all unless some event happens..
It would be an interesting test - but beware that there are multiple approaches, and even plain ol' definitions, for how a 'game engine' differentiates from an 'operating system'.
The two systems of thought have been intertwined, commercially, for decades...
I would rather see how they fare with a classic GUI like a text or spreadsheet editor. Maybe for simplicity sake a calculator or an IRC client, then you just have to join an active channel and let it run for a while.
