We were using the admin account which has full permissions. I think it's actually an issue with that version of the 'OpsGenie for Splunk' app. It works on the latest version of that app, 1.1.6, which was released recently.
The functionality on a cluster is still a bit odd, you have to modify your URL to
/en-US/manager/opsgenie/apps/local/opsgenie/setup?action=edit
to be able to access the app setup to enter an API key as the link doesn't appear via the GUI. The users that create the Splunk alerts also need to have the "list_storage_passwords" capability, as you mentioned.
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