Sure seems very cumbersome. I mean, you will need a power outlet for the PS2, so that means you aren't doing it on trains or in cars or something like that. So that must mean you are only looking at playing while you are in hotels or staying at houses, which already have a tv you could use. If you're in hotels, most have the connectors to hook up a PS2, or you could buy the coaxial adapter for the most basic tv's.
For how much you're going to spend for the elgato tv thing, you might consider buying a PSP.