Hi, I have been working with G Suite for over two years now and all migrations/sync via IMAP, POP, G Suite Sync for Microsoft Outlook have been perfected. I believe I know the root cause of the error and lack of syncing between the G Suite account and your Outlook profile. I would be happy to assist you in a timely manner (up to 1 day depending on which of the above options you are currently using and due to which the synchronisation is failing).
Please, do let me know further details via private messaging and would be glad to assist you on the matter. My goal is to ease the process for you and reach desired resolution in a timely manner as I know from personal standpoint how important the email delivery and sending is to a business.