Get a seat. The bassinets are small; I don't think they're designed for a 1-2 year old. I've used them on a 767-400 bulkhead with a three month old and she used >50% of the space. I believe the sCO 777-200s in BF have the flip-up thing above the TVs that looks even smaller, in my opinion.

Additionally, you're not supposed to use them when the seatbelt sign is on, which can be for hours, and if the FAs enforce it every time it comes on and off, you'll be taking the kid in and out constantly, which wakes them up and is a hassle.