How to set BLE client name?

Advertising wastes very little power, for some devices (beacons) it is the only way to do BLE and they can still last year from CR2032 coin battery. With connection it is said to allow to use even less power but it is all matter of settings (connection interval, advertising interval) so if advertising is good enough for what you want I wouldn't bother with connections and GATT.

For power draw it does not matter which side of connection it is. Scanning for advertising packets however takes more power, not sure if android phones can somehow optimize this case but with espruino it needs radio to be constantly on for long time (seconds) to listen for incoming packets. So for two battery powered espruino devices connection is better.

I envisioned it something like a standard BT pairing.

Yes it can work like that. The natural way is to make espruino device to be gatt server (= peripheral) and phone to be central/client - phone will scan for device when it is advertising (by name or anything else) and connect to it.

As for pairing there is such a thing called bonding which even allows secure encrypted connections.