Do double check that your daughter will be less than 24 months (from birth) to be eligible for the infant fare. (Also, if you aren't used to long flights with a large lap child, it might be worth thinking about whether you want to buy her a seat anyhow; we decided 9 months was the cutoff for our kids, after a very unpleasant flight, which was mercifully only two hours.)

I had some BR award space a while ago on 12/20, but it looks like there just isn't any in the couple weeks before Christmas available now…it's been snapped up. In that period, you can take UA standard awards to other points in Asia, and then connect to BR flights, so it doesn't look like there's a blanket blackout in effect.

You are looking for a particular flight at a peak time of year, for several people, and that can be too high an expectation for saver awards. Explore some alternative routings: award tickets do not need to be all on the same carrier. One way to do this is to search for one ticket at a time, and then explore where there is availability before looking for 3 on the same flights.