I would not put a 3-year-old on a plane without a carseat. The law doesn't care, but the laws of physics do. (An actual carseat. Boosters are not allowed on planes and I would not put a child under 6 in a booster).
Children can be easily killed by impact on the ground (that would be survivable in a carseat) or turbulence. I don't have the link but I watched a video of moms whose babies died in plane turbulence or accidents (where most adults survived), and these lap babies just flew. So sad. An active 2-year-old won't just sit in a lap belt the whole flight either, w/o you going crazy. :) Bring that carseat. And you'll need it for cars in Europe, too, and to get to the airport.
We bought the Cosco Scenera seat ($40) and rear-faced the kids on planes until age 3, then forward-faced them until 6 in the Evenflo Maestro (also a cheap, light seat, but only for FF). Both install easily on a plane or rental car.
An active toddler on a plane? Good luck! I've been there and can't imagine how awful it would have been with him taking his belt on and off and bouncing around. The carseat was a lifesaver for sanity, and the kids slept (and safety, of course). Have a fun trip. Europe sounds great!
Cares harness (a good choice too) goes to 44 lbs., I think? Not sure, but for some kids at age 2 or 3, the limit is too low.