How Do I Grant Full Access to an Office 365 mailbox without it also loading into Outlook?

Yup - you just need to disable AutoMapping on the mailbox. But, it's not as easy as flipping a switch. You need to remove the current permissions and then re-add it with the automapping property to off.

So, if your account is the admin account and you need to have access to sharedmailbox you need to execute this (you must connect to Exchange Online Power Shell first):

Remove-MailboxPermission -Identity [email protected] -User [email protected] -AccessRights FullAccess

Add-MailboxPermission -Identity [email protected] -User [email protected] -AccessRights FullAccess -AutoMapping:$false

Note:

You can also change your outlook settings so shared mailbox is not downloaded. It's under your account settings --> Advanced settings.