[Spicymoose]

Not only does checking prevent you from getting check/raised, but it also induces a bluff by him on the river, which he will probaly take. If he is behind, he may not call your turn bet, but will give you the money on the river. If you are behind, there is no reason to bet the turn. On top of all this, if you improve, you can pop him on the river.

Edit: I guess an argument for betting could be betting as a form of inducing a bluff. If he will go for the double check raise with air, betting could allow you to get a lot more of his money when he is behind, and since you have so many outs, it wouldn't hurt all that much if you were behind.

I still think checking is probably best though.