SharePlay not reliably playing to Apple TV

I’ve been trying to have some TV nights with a long-distance family member, and it’s been a frustrating experience. We start with a FaceTime Audio call, then I start SharePlay on my Apple TV, confirming on my iPhone. But on her end, the show starts playing on her iPhone, and her Apple TV is unavailable as an Airplay receiver.

If we hang up the call, and she starts playing a show on her phone, the Apple TV will become available as an Airplay receiver and she can send what she’s watching there. Then, she can pause, I can call her back, and if I start SharePlay again, it’ll play on her TV. But if we switch shows, her Airplay functionality will break, and it’ll begin playing on her phone again.

Has anyone seen similar behavior — or better yet, solved it?

I doubt this matters, but in case it does, she’s on my Apple One family plan.