At the moment, when we run a GET REST API to get a request type details, the response will be shown like below:
From here, we could not actually find the key of the request type that is being used for JIRA when storing the Customer Request Type Custom field value in the database.
To include the key in the response or better adding the lowercase project key at the front as how it is in the customfieldvalue table.
See below KB article for a workaround. However, please take into account that this is not supported and it may stop working without any previous notice: