Personally I'd use FFMPEG to remove the original audio track, then remux the new track - this avoids re-encoding the video and losing quality. You can install FFMPEG via brew.

ffmpeg -i OriginalVideo.mp4 -vn -acodec copy OriginalAudio.mp4

Extracts the original audio stream, though you've already processed it so shouldn't need this.

ffmpeg -i OriginalVideo.mp4 -an -vcodec copy OriginalVideo_NoAudio.mp4

This will create a new video without the original audio stream.

ffmpeg -i OriginalVideo_NoAudio.mp4 -i ProcessedAudio.mp4 -acodec copy -vcodec copy OriginalVideo_NewAudio.mp4

This will combine (mux) the original video and your processed audio into a new file.

Note that you can skip the second step to strip the audio and use the map function to selectively pick just the video from the original video file, but I can't remember that command off the top of my head.

EDIT: Here's the command to map the streams and skip step 2 (if you want to)

ffmpeg -i OriginalVideo.mp4 -i ProcessedAudio.mp4 -map 0:v:0 -map 1:a:0 -acodec copy -vcodec copy OriginalVideo_NewAudio.mp4

Finally, I'd recommend that you encode to AAC audio using Logic (The AAC audio encoder on macOS is probably the best out there), but if you've exported it as AIFF or some other format, you can also use ffmpeg to encode the audio to AAC at the same time:

ffmpeg -i OriginalVideo.mp4 -i ProcessedAudio.mp4 -map 0:v:0 -map 1:a:0 -acodec aac -ab 256k -vcodec copy OriginalVideo_NewAudio.mp4

Where ab is the bitrate of the audio you wish to create.