You might be able to get this done with the Extension for JIRA Service Desk add-on. It offers the possibility to calculate an SLA based on a date (such as due date) and it allows you to show the SLA timer on the customer portal.
The add-on is commercial. So if you are looking for a workaround, you can always mention the due date in a comment to your customer as soon as you decide to set one.
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