What sort of events play live renditions of the Star Spangled Banner, that are preferably free for public? Sports events come to mind, but it is beginning to be late to get good tickets for cheap.
We are staying in NYC for a week in late September.
As you said, sporting events are usually your best bet for a guaranteed National Anthem. While they (usually) aren't free, if you're not picky about your sport, there's probably more than a few that you can get for relatively cheap.
You can also check out minor league baseball, but that will be over by the end of September.
Major League Baseball can be very hit or miss as far as ticket prices go at that time of year. There are almost aways some very cheap tickets available if you're willing to sit in bad seats, but that's up to you.
Beyond the professional ranks, you can always check out some younger teams. For instance, a high school football game will also play the National Anthem before the game. That's starting to get pretty far down the list of places you probably want to end up just to hear something you can get on your phone for less than $1.