Even when with a lap child infant ticket that is 10% of (some) fare, the tickets are so often messed up in such a way that it's an uncommon opportunity to enter in a frequent flyer number for the lap child infant and getting it to stick without it costing accompanying ticketed caretaker elite status benefits and/or miles.

While I'm sure mess-ups do occur, it's not frequently that major alliance airlines grant credit and/or their programs grant credit in such circumstances.

... at least there is the possibility of getting points for the ticket charge itself, but beyond that it is all a (generally non-winning) gamble.