Hi,
HikaShop has no problem sending emails via Amazon SES.
The invoice plugin will add an attachment to the emails.
Amazon SES has limitations on email size
repost.aws/questions/QUyA0h1vpsSgqMtSgHP...achments-above-10-mb
but that should not be a problem.
Try disabling the plugin and see if you receive the email when you change the status.
If, even with the plugin disabled, it doesn't work, it's likely a problem with the settings under the Emails section of the Main tab of the HikaShop configuration page. Please review them and make sure they match what Amazon SES requires.