On a Bark Phone, you can use Android wearables (like Fitbit, Pixel Watch, or Galaxy Watch) via Bluetooth/Wi-Fi, but Bark Phone data plans do not currently support using the cellular features on those wearables.
In other words, Bark Phone data plans cannot add an eSIM to a Pixel Watch LTE nor a Galaxy Watch LTE.
If you are not subscribed to a Bark Phone plan, contact your wireless carrier (e.g. AT&T, T-Mobile, Verizon) if you have questions about their support of cellular features on LTE smartwatches.
Smartwatch app not loading when Bark is installed?
Sounds like you may have accidentally blocked access to the smartwatch app(s) through your screen time rules. Here's how to allow apps and sites on Androids, iPhones, and Bark Phones filtered by the Bark Kids app or the Bark Home device.