At the moment, this is not possible to do with Jira Service Desk. At least not with the native code for Jira. There is an existing feature request open for this feature in JSDCLOUD-3432.
In the meantime, this kind of customization to notifications might be something that you could accomplish with a third party plugin for Jira. From a quick search of the Marketplace, I think that you might be able to customize these notifications with a plugin like Jirassimo.
I hope this helps,
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