Welcome to FT!

This happens all the time with parents who are meeting the child, for whatever reason, at the destination (such as the child already being there with the other parent, as in your case). You shouldn't have a problem. I wouldn't even expect to be grilled about the reason, though you have a perfectly good one if anyone asks.

As to whether it will be free: that depends on your free checked baggage allowance. That, in turn, depends on the fare you paid and on your United or partner elite status, if any. We don't have enough information to say one way or the other.

All that said, the definitive answers can only come from United. The best FT advice will be found in the United forum. I've asked a moderator to move this thread there.