If you need a really high resolution video, I recommend using Screenflow. Screen flow can record uncompressed video, and export very high resolution video.
That being said, you might not notice quality problems with Hype’s standard mp4 export – I recommend testing it out. After Effects can ingest the MP4 format.
If you are exporting as HD (1080p), you should be fine with the 1080 lines of resolution in your 1440x1080 document. I recommend doing a test and seeing if the quality is suitable for your needs.
Keep in mind that you can always slow down your animations, and then speed them up later. In a browser, you might not be able to get a full 30FPS, but if your animation is twice as long, you can compress time and bump up the frame rate.