Metric alerts do not support customizing the json payload. In this cases best is to have signal that starts Azure Automation runbook, Azure Function or Azure Logic App. In one of those you can process the payload to your own liking and send it to the destination you want.
With that said I believe you can add custom properties to the already defined payload by using properties.actions.webHookProperties. . That allows you to add additional name value pairs to be passed with the action. You will have to test where those appear in the alert schema though.
P.S. Do not forget to mark this reply as answer.