Apple Watch does not show up in iPhone Bluetooth list. Will not pair

Its not supposed to appear in the BT list.

Pairing is done through the Apple Watch App, it will never show up in the bluetooth settings list.

Follow the steps here: Set up and pair Apple Watch with iPhone - Apple Support

While the pairing is done in the Watch app on the iPhone, my Apple Watch still appears in the list of Bluetooth devices on my iPhone.

Really? Never seen one show up there. But I'll defer to you. I no longer have my watch so can't say for sure.