Unfortunately there is no way of doing this without building your own service that hits or is triggered by the DB. The current JIRA Service Desk API does not have any support for accessing customer satisfaction. You could query the DB and use a batch load to your CRM or create your own service to pull the information you are looking for until Atlassian adds customer satisfaction to their API.
You can also check the marketplace as there are a number of add-ons for surveys that may provide the functionality you are seeking.
The following DB query will give you all issues (and the issues project) that have a Satisfaction Comment added.
select p.pkey, i.reporter, ep.json_value from entity_property ep
join jiraissue i
on (i.id = ep.entity_id)
join project p
on (i.project = p.id)
where ep.property_key = 'service-request-feedback-comment'
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