We flew with our two girls (4 and 2 1/2) to LA recently. Of course, I had to buy a ticket for each of them, but as it became apparent that the flight was becoming full, the flight attendants came over and asked if my youngest daughter was a "lap child" or if I'd purchased a ticket for her. I told them I had a ticket and offered to retrieve it but they said it wasn't necessary. My daughter is very big for her age, sometimes people think she's 4, so I was surprised that I might have been able to pass her off as under 2.
I've flown with my girls many, many times before they were 2 and they've always flown as "lap children". Fortunately, we've never had a problem. Sure, they're a little easier to handle when they are in their own seats, but taking them along without a ticket was "doable" as well. I understand what you mean about paying for a ticket if your child has literally just turned 2 and I have to say, even though I'm sure it won't be a popular opinion here, that I would probably just try to get by without the ticket. Just my 2 cents!