I'm creating a music video where the only audio is a single MP3 file. From what I can tell, all rendering profiles would pass the audio through an encoder, which would lead to some unnecessary audio degradation.

Is there any way to avoid this and just use the existing MP3 audio file in the final video file?

Thanks.

yes, Google remux or multiplex. Tools like ffmpeg, tsmuxer, mkvmergeGUI, MP4Box are suitable depending on codec and container combinations required.