No, you can't lie about her age; they will look at her passport and know exactly how old she is. I don't fly DL, but you probably have to call to ticket a lap child. I haven't seen this mentioned yet, but she isn't going to fly free, even if you hold her in your lap. You certainly won't be getting something for nothing; far from it. This is the same for almost every other airline. From DL's webpage:
International Taxes & Fees—
For infants under the age of two and held in the adult's lap, the cost is usually about 10% of the adult fare plus any international taxes and surcharges, which can be significant.

Also, your child is pretty close to the age cutoff. If she turns two during the trip, you're going to have to purchase her a ticket for the return.

If you have purchased a seat, and she is in her carseat, you do not have to take her out and hold her during take off and landing on a DL flight. That makes no sense. You may make your own choice, but I have to agree; do not count on having an empty seat in which to place your child if you haven't purchased one. I do not understand your sitter's story because IME, flights to Lima are packed. I find it really strange that she was moved up to first with a lap child when she had a coach ticket.

I think it quite difficult to manage a lap child of this age on such a long trip.