I've deliberately turned off all the permissions I don't want, but I keep getting a pop up that Garmin is requesting permission. How to stop that?

What do you mean by pop up? Is it a notification from the connect app? 

Hi, yes it is a Garmin Connect notification pop up.

If you are on a somehow recent android version you can long press any notification and disable it. 

But keep in mind that some functions will not work properly if you do not grant these permission. 

So there's no way to suppress the Garmin notifications at the source?

You can disable push notifications completely in the Garmin connect app settings 

Thank you!