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.