I understand that you are wanting to be able to use a Jira Service Desk customer account in order to be able to access and edit Entity properties via REST.
This is not possible to do with those specific account types. The reason for this is explained in some more details in the Jira Server Developer documentation: Entity properties. From that page
To modify, add, or remove the properties, the user who executes the request must have permission to edit the entity. For example, to add new property to issue ENPR-4, you need permission to edit the issue. To retrieve a property, the user must have read permissions for the entity.
Jira Service Desk customers are not licensed Jira users. As such they don't use a license seat, so you can have an unlimited number of them. However without at least first having application access, which in Jira Server is granted by having membership in a group that has that application access, such users can't even begin to see those issues in Jira itself. They would also then need to meet the permission schemes requirements to be able to edit the issue itself.
Jira Service Desk affords unlicensed users in the customer role the ability to create requests, see their requests, comment, etc. However the ability of those users to do that is restricted to only be possible within the customer portal, via the Jira Service Desk mail handler, or via the Jira Service Desk REST API. All of these methods though are meant for the customers to interact with the request, but not the issue itself in Jira. More on this distinction between Issues vs. Requests in Jira Service Desk: Issues vs. Requests.
Sorry this is probably not the answer you were looking for, but I at least hope this helps to explain the behavior here.
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