You can pop it anytime after a song is sung. Just make sure that you have enough MP to make use of all 30 seconds of the boost.

Paeon + Voice is great for bringing your party's TP back up, along with Ballad being great for restoring MP when your healers need the resource after reviving a couple of players. Reqiuem + Voice is great for a party with strong casters, and can be great for pushing phases as long as your party is not in trouble with MP/TP management.

If anything, a Battle Voiced song does not affect the BRD singing it. With Paeon, a BRD would have to play it for 15s longer since they do not receive the boost from the song buff. Again, Battle Voiced songs do not affect the BRD singing the song with their cooldown active.