If your audio is out of sync, you can fix using ffmpeg.
ffmpeg -i source.mp4 \ -i livestream.mp4 \ -map 0:0 -map 1:1 \ -acodec copy \ -vcodec copy \ -itsoffset 0.100 \ output.mp4
The ffmpeg fix is fast. A 1GB file took less than 30 seconds to run. Adjust -tsoffset until you find the sweet spot.