Another Update. More things I tried:
Move theme files to a safe location and pull entire site from develop including DB.
Disconnect site, delete site, remove local application, remove local sites folder, remove local folder from library: https://localwp.com/help-docs/troubleshooting/uninstalling-local/
Redownload, reinstall local, reconnect with new API key to WP Engine.
Test that file changes are detected by theme from pull - they are! YAY!
Override existing theme files with reserved theme files.
Test that file changes are being detected - Yes so far! YAY success!
I found this article about caching issues for another user that is very similar to my problem, but they were able to confirm it was an issue with their theme. However, I am not using the same theme and the problem seems to be resolved without any change to my theme files. I am fairly certain that the issue was somewhere in Local caching, but I was unable to discover it.
I will keep this post updated if there are any further developments to share. Thank you for reading 