However, when the device is not connectable, the advertisements will have data type ADV_NONCONN_IND. That basically means that the device doesn't ever listen for incoming data, so the central will also never request a scan response, and the local name will never be broadcast.
In this case, it would be better to use ADV_SCAN_IND instead; I've already verified that the name is shown again in this case.
In BLEPeripheral.c, the local name is put into the scan response data block: https://github.com/sandeepmistry/arduino-BLEPeripheral/blob/master/src/BLEPeripheral.cpp#L121-L132
However, when the device is not connectable, the advertisements will have data type ADV_NONCONN_IND. That basically means that the device doesn't ever listen for incoming data, so the central will also never request a scan response, and the local name will never be broadcast.
In this case, it would be better to use ADV_SCAN_IND instead; I've already verified that the name is shown again in this case.