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that why I'm trying to solve it anyhow by (like You said) c in Arduino envoirment.
It is perfectly OK to use any solution. It is just that we cannot help you much here with details of using BLE in Arduino.
This is my first tries with BT/BLE.
We can still discuss generic BLE concepts (advertising vs connection) and how to do the Bangle.js side here. For that you already had some suggestions.
Advertising is short data packet broadcasted periodically containing typically a device name but you can add anything small there (up to 31 bytes total or 2*31bytes - with optional advertising response packet). Then just scanning on the other side and parsing the advertising result should be enough - inside your MyAdvertisedDeviceCallbacks::onResult(BLEAdvertisedDevice advertisedDevice).
Currently it checks serviceUUID which can be there too
advertisedDevice.haveServiceUUID() && advertisedDevice.isAdvertisingService(serviceUUID)
but there can be also other custom data you put there.
I'm not a bot (I know, all bot can say it :))
I'm not familiar with JS, that why I'm trying to solve it anyhow by (like You said) c in Arduino envoirment. This is my first tries with BT/BLE.