Why not 1440p with integer 2x scaling?
Native 720p does look slightly better than integer scaling to 2x because of subpixel stuff, and generally liking that LCD look. But integer scaling would be enough really.
Sounds like a fun project, you should build one.
Taking in a 720p hdmi signal and pixel doubling to 1440p or tripling to 4k and outputting that as a valid HDMI signal without adding more than a frame or two of lag would require custom chips. Although maybe a software solution involving a gpu might be possible?
Implementation is left an as exercise for the reader.