Yeah, it's definitely a video call routing issue with Kids Messenger. No reply from Facebook so I did some further testing. Voice calls actually work via BT and wired headphones, until you change to video, this then forces it out the iPad speakers, and not the headphones. Once video call is selected, and you change back to voice call, interestingly it does not route back out to headphones, instead stays via device speakers. When you disconnect the call, you can actually hear a click in the headphones, so they are activating, just not routing the video call correctly.
Noticed there was an update of Kids Messenger waiting in App Store -> version 227.0. Downloaded and installed, tested, still same issue. Also noticed an IOS update -> 16.3. Downloaded, installed, rebooted iPad, waited with hopeful patience, and ... no change, same issue :(
There's no bug report feature for Facebook however you can supply feedback from the following page -> https://www.facebook.com/help/messenger-app/213724335832452/
It would be great if anyone/everyone to submit feedback to Facebook, either via the "Was this helpful" widget and selecting No, or via the Feedback Form -> https://www.facebook.com/help/contact/268228883256323 as the more people that bring light to the problem, the quicker the fix.