Suggestion: 

Currently EPIC Status is not an available field in Edit Issue Action in Automation for Jira.

Make EPIC status as an available field in Edit issue field list. 

 Workaround

Copied from comment below:
The work-around for this is to use advanced edit for the custom field (epic status), such as:

{
    "fields": {
        "customfield_10010":

{             "value": "Done"         }
    }
}