The name is placed in the advertising data automatically when you call bt_le_adv_start(BT_LE_ADV_CONN_NAME). When you call bt_set_name(), that is updated as well:
That said, the name is placed in the scan response, so you need to make sure the scanner is performing active scanning.
From: j.cheng via Lists.Zephyrproject.Org <j.cheng=de-haardt.com@...>
Sent: 10 February 2020 10:56
To: Cufi; Cufi, Carles <Carles.Cufi@...>
Subject: Private: Re: [Zephyr-users] bt_set_name() and setting local name #ble #bt #bluetooth #api