Jailbreak and install display recorder. Profit.

Unfortunately, from what I've read, the image is somewhat degraded -- you won't be getting Retina-level resolution.

Reflection has recording built in. It was pretty laggy for me, but my MacBook was on wifi and maybe bogged down with other stuff running and the app I was running pushed a lot of pixels. (That's a whole lot of 'but' there. I need to try it again.)

No question. I'd say the biggest issue there would be artifacts and frame rate. But jailbreaking is really the only other option, so I think it's not a bad plan.