The best way to save energy would be to use SLEEP_MODE_DEEP in between button presses. This would shut everything down and draw as little power as possible. When the user presses a button, bluz can wake-up and start advertising, send whatever data it needs to send, and then go back to sleep.
This would have more lag then if it was always connected. If bluz needs to wake up and reconnect each time, then it would take a little longer. However, it may not be significant for your application so it may not be a big deal.