Power BI API : GET Refresh works, but POST does not

I was trying to save you time but it seems like you insist on using Postman. So please verify this

https://nonodename.com/post/pbirestapi/#:~:text=First%20of%20all%2C%20got%20to,can%20be%20plugged%20into%20Postman.&text=Hit%20%E2%80%9CGet%20New%20Access%20Token,login%20and%20get%20a%20token.

Then verify that you added the App registration to a security group and at the tenant level

https://learn.microsoft.com/en-us/fabric/admin/service-admin-portal-admin-api-settings#enhance-admin-apis-responses-with-detailed-metadata

You should also know that a new token is generated every hour and you can only use POST https://api.powerbi.com/v1.0/myorg/datasets/xxxxx/refreshes up to 8 times a day for a shared workspace capacity, just like you schedule a refresh in the service.

So if you are planning on using retool.com then you know what to expect. 

However I would recommend that you use Power Automate to create a button in your reports for an unlimitted on-demand refreshes.

Here you go

Trigger a cloud flow from any Power BI report - Power Automate | Microsoft Learn