You may exile and return Brago using its own ability.

Brago's last ability exiles and returns all the targets during the combat damage step, after combat damage is dealt. You can't target any creature that didn't survive combat.

If you exile an Aura with Brago's last ability, the Aura's owner chooses what it will enchant as it comes back onto the battlefield. An Aura put onto the battlefield this way doesn't target anything (so it could be attached to an opponent's permanent with hexproof, for example), but the Aura's enchant ability restricts what it can be attached to. The Aura can't enter the battlefield enchanting a permanent that enters the battlefield at the same time. If the Aura can't legally be attached to anything, it remains exiled.