Hello aldicins,
First off, please make sure that these are the steps you are following to set this up: Take a Live Photo in FaceTime on iPhone. If those are the steps you are taking, and the issue persists beyond the troubleshooting steps you've already taken, further assistance may be needed. We want to make sure you reach the right team for help with your iPhone and this FaceTime issue, so we recommend that you contact Apple Support directly. Depending on your region, you may be able to reach them here: Get Support - Apple.
Thank you for participating in the Apple Support Communities and have a wonderful day.