Okay, I was able to reproduce the problem with your files. The problem seems to be with the video, not the audio file, because the problem occurs when I pair the video with any other audio file, but it doesn't occur when I pair the audio file with some of my videos. However, it does occur with some other of my videos.
It seems that when some videos load, they reset the iOS audio engine and unload the recording. I tried changing the sequence so the video starts first, then the recording starts, but that didn't help because it takes longer for the video to load.
One way you could work around this is to record an automation track for each song that starts the video, then 2-3 seconds later starts the recording. Then when you select a song, start the automation track rather than starting the video and recording separately.
Alternatively, you can turn on Help > Utilities > Use Old Audio Engine and then the video won't interfere with the recording. With the old audio engine, tempo clicks won't be as accurate and you won't be able to use multi-route output or inter-app audio.