Dear Julian,
Thanks for your updates here.
JSON code may be difficult to meet your requirement. The feasible way in your environment may be to add three buttons in the item form customized with Power Apps and then trigger the three flows by the buttons in Power apps.
Here is one of the demo flow for your reference. The Send the email action is triggered by the button in Power apps.
You can type flow1.Run(DataCardValue1.Text,DataCardValue3.Text) to trigger the flow and do the same things for the left two flows as you like in your environment.
Result:
Welcome to share any updates when you have time if you need further help on this issue.
Sincerely,
Cliff | Microsoft Community Moderator