Overcome API KEY request at Relationship edit / delete for users of certain Drupal Role
you seem to me mixing two related but different concepts:


the REST API, that's an interface to connect to Civi, mostly from external servers (eg another application). In principle, the api_keys and server key shouldn't be visible to the end user but stored in a configuration setting somewhere.
the ajax API, that's how a user interact with civicrm to avoid to reload the page completely. This one doesn't need the api_key, as it relies on the user authentication to know what is allowed


In both case, it's not possible to bypass the permissions from an external call directly, it would defy the purpose of having a permission system in the first place ;)

What I would suggest you to do is to write a simple civicrm extension that offers a "member.resign" API and a new "quiter" permission (name to be improved ;) and configure the api so it's only the quiters that can resign.

your api can then call the apis to remove the relationship or whatever else you want to do, and can have the skip_permissions parameter

Then you simply have to grant that permissions to the roles you want in drupal and call your new api CRM.api3('member','resign', {...}) and voila

As an aside, I'm not a lawyer and I don't want to speculate what you do with your contacts that resign, but marking them as deceased is likely to be considered illegal in most jurisdictions ;)