I think if you upgrade to C, for your infant child (to have him/her on your lap) you'll have to pay 10% of C fare.
Nope, you just pay 10% of the economy fare - I have done this. The problem is a lap child on a miles ticket, then you do end up paying 10% of the C fare.
If travelling with an infant on an overnight flight, I have found it best to take the bulkhed row. The crew have 'cribs' they can attach to the bulkhead for the infant to sleep in. The infant sleeps much better this way than in a seat.