It is rare to be able to book three F award tickets on the same flight, regardless of the points currency.

If you redeem via Asiamiles, you will be able to waitlist for an award. You CANNOT do this with AAdvantage or Avios. Main/Only benefit of using Asiamiles is that you can waitlist on more than one flight on the same day. Downside with Avios is that they charge you for changing award flights (plus they charge fuel surcharges). AA miles are clearly the cheapest and pretty flexible too.

Of course, using Asiamiles costs more points and also results in hefty fuel surcharges.

Personally, depending on how old your "little one" is, one F and two J on the same flight makes more sense to me over two F and one F on different flights. Your little one is not going to care whether s/he is in Y, J or F. It is really the parents who care, and personally, I would rather travel as a family than be separated but flying F.

As for whether you can change an award flight on the same day if availability opens up, this is an unlikely scenario especially if you expect to do this at the airport. I just wouldn't count on it.