Can I render the GUI in 1440p?

I'm fairly sure the answer is no. I've just upgraded to a large 1440p monitor for both my PC and the Shield to use in my small room at university. The Windows PC is obviously perfectly happy to output at any resolution I want, and even prompts me if I use anything other than the display's native resolution. The Shield on the other hand has extremely few settings in every department, including HDMI. I get that Google is trying to simplify their various software for the average user but come on, no option to set the output resolution? Ridiculous. Under Settings, HDMI I've set 'HDMI optimisation' to 'best resolution' mode (that's the only relevant setting I can find). I've then restarted and it still appears that the Shield is outputting 1080p and letting my monitor scale it up (crudely). I'm not only going by my judgement of the image quality. When I take a screenshot it saves as 1080p.
Presumably people with 4K displays aren't experiencing such a problem? The device is, after all, supposed to be 4K60 capable.
I haven't tried any custom ROMs (nor do any current ROMs offer a 10-foot UI as far as I'm aware), I wonder if it would help though?
Nor have I looked into using the Xposed framework (which I've only ever read about and never properly used), though I'm doubtful whether that could help.
My problem isn't with 1080p HD itself. As far as video is concerned I can't even tell much difference between 720p and 1080p at a normal movie viewing distance. It's when any GUI renders at non-native resolution that it becomes really obvious and annoying to me.