This is such a stupid thread as it ignores that the latest Vita model already gets acceptable battery life in the 2013 model while outputting a resolution more than twice that of the 3DS. By 2016 I'm pretty sure Nintendo should be able to pull something off that's more power that has at least it's battery life. Even the games which miss full resolution are still running at twice the resolution of the 3DS, so I'm not sure why you bring it up as a negative in the OP. If they can't manage 1080P and a large screen fine, but we've already seen that there are better alternative routes Nintendo can take but probably won't, and it's time people stop making excuses for them as they never pass their savings on to the consumer anyway.
Vita's resolution isn't more than double the 3DS, it's slightly less than double. The Vita is/was also not a profitable system.
That doesn't account for tech developments since then obviously. But we're looking at Vita or iPod Touch level specs as a target for a cheap system and not beyond that really.