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Hi Micah,
Thanks for posting in the community.
After searching, here is a thread where the customers discussed a similar requirement. You may need to use SharePoint Framework to upload the custom CSS file to SharePoint to hide the Sync button.
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As the issue is related to customize the SharePoint site and we have a specific channel Microsoft Q&A Community coping with such customizing site requirements, I suggest you post a new thread there for expert help.
Sorry for that our category may have limited resources on checking on the issue further.
Thanks for your understanding. George