To throw down the gauntlet means to issue a challenge, and in this context the speaker is saying that if the baby challenges them, they will be happy to accept.
Consider the staetement "Oh, baby, we both know that if you challenge me I 'll be only too happy to pick it up.". Based on this statement, is "Oh, baby, we both know that if you throw down the gauntlet I 'll be only too happy to pick it up." a true statement? Yes/no/unknown