Turns out it also has the most horrific coil whine I've ever heard, think I'm just going to return it. Would the performance issues be related to the coil whine (I say whine, it's more an incessant scream)? Is it just a faulty card, rather than a software problem as I previously suspected?
Because I know how frustrating it is to look for an issue and the person who found the answer doesn't share; the coil whine issue was fixed by limiting the FPS in Wattman. The higher the FPS, the more it screams. Still no fix for the low performance, but interestingly, it'll run Grand Theft Auto 5 at shockingly high framerates, so much so it induced the coil whine issue that I hadn't seen outside of low end benchmarking software.
I do wonder if the card is stuck in a low usage state when it runs undemanding titles. GPUz shows it's working at max clock speed, but the power draw is low, around 37 watts, and the temperature stays minimal. Since it looks more and more like a software problem, I doubt I'll get away with returning the card *sigh*.
Success! My TV had an HDMI overscan feature, which I didn't realise. I've always adjusted that with HDMI scaling in AMD settings, setting the adjustment to 4%. Since disabling overscan on the TV and resetting the card to 0%, games are running fine. 50FPS in Skyrim (capped) and 30FPS in Age of Empires (also capped).
Hope this helps someone, I see it's a common issue on these forums.