We use Aha for both Feature Requests and Defects and need to be able to designate if the JIRA issue type is an Epic, Story or Bug when linking to JIRA.
The best way to achieve this today is to would be to configure multiple JIRA integrations. Each integration would be separately linked to Epics, stories or bugs.
By putting this in place, this would allow you to create features and defects in Aha! and designate them in JIRA based on the issue type of interest.