HDCP problems with apple tv

I mean getting another HDMI cable to test. You’ve simply tried on different HDMI ports on the tv.

If you have a small black box then it’s at least a 2nd Gen. You can go to settings - about to see what firmware it is running.

Make sure HDMI cable is secure. Try a different cable and port. Test on another TV

iMovie content is personal and not copy protected

No. HDCP is a copy protection that is in place when streaming commercial content only. So that fact that iMovie works wouldn't be relevant.

It could be, in order to rule it out you would need to try another cable.

Out of curiosity what model Apple TV are you using?

Identify your Apple TV model - Apple Support