ble<=>ip

Is the BLE radio in the Bangle very short range?

It's not amazing I'm afraid - I think the case gets in the way, and the aerial isn't as good as the Puck/Pixl ones.

It is paired with my PC but I presume there is a difference between connected/disconnected and if I disconnect from the browser it should become available.

It depends - it's worth checking in the OS if you had something like HID enabled the OS will tend to keep the connection open. If you did have HID enabled in the Bangle settings that could explain a lot of the instability connecting though...

my goal is for IP (4 or 6 don't matter for this purpose) over BLE

I'd be very interested in this. It actually struck me that you could use a similar method to MQTToBLE to do something like an AT-command interface to a bridge. It's a bit hacky but at least it's all 'off the shelf'.