they could pixel double to 2k and then upscale and it would still look better than 1080p

It most definitely would require a CPU upgrade. The IPC is just too low to keep up with a GPU trying to render an image at 2160p. The bottleneck would be too great. The full R7 Graphics on the die in the ps4 right now is already being bottlenecked by the outdated and underwhelming architecture.