What about sideloading the actual Audible Android app (the one from the Google Play store) to the device and using that instead of the Fire's integrated Audible? With that app you should be able to log in and out of various accounts. Fire's are designed to be integrated with your Amazon account for consuming Amazon content (which includes Audible) so AFAIK there is no way to work around it and access an Audible account that's not linked to an Amazon account.
EDIT: Yeah, from Audible's help area. Audible on a Fire requires a merged Audible/Amazon account...
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Note: If you are signing in to your Audible account with a unique username and password as opposed to your Amazon account, you will need to merge in order to see your Audible audiobooks on your Amazon Fire
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