Update 19th April 2021
Thanks for your patience. We are continuing to fix email-related issues in Jira Service Management. We are exploring ways of making this possible but no work is planned on this in the next six months.
Jason and the Jira Service Management Team
At the moment, it is not possible to customize the Service Desk mail handler. When you try, the following error will appear
The email handler Service Desk Mail Handler can only be used by JIRA Service Desk. Please choose a different email handler type.
It would be better to be able to configure the mail handler like the normal JIRA mail handler for some options (regex, Bulk, etc) which will affect the emails being processed by the mail handler.