A long time ago I used an ipad 2 for this, It was mirrored through the 30 pin but its output was only 720p- the video looked fine and was smooth.
I think the airplay mirror seems the best way to go and a better network may give you a better signal. There is a difference in latency when capturing with the blackmagic SDK vs quicktime, it is a little bit faster, but depending on your machine 720p should not be a problem. I would try isolate the issue, use another source for your capturing (not the ipad via airplay) and test the quality and latency. If this is good you know your capture is good and it is the source. Next try different network configurations- do you have access to a wireless AC router, this is very fast and high bandwidth. Last, the apple TV is a cool little device, but just to check, try do the airplay mirror with a laptop with a good GFX card, maybe the GFX processing on the apple tv needed to get the compressed signal from wifi, decompress it and display it is too much work. Fred