I’ll leave it open to the community to comment on the use of Pushover on a Fire 7 with Google Play sideloaded…
For what it’s worth, I was able to get Pushbullet working on a Fire HD 8 (10th generation), but the same thing happened with Pushover for me where it let me sign in, but then failed to register the device.
As noted in my reply to your comment in the other post, since this is a Amazon Fire Tablet, if you’re using Fully Kiosk Browser to display a dashboard, you can also use that to play sounds on the tablet.