I've used the Telex 850 ANR for nearly 10 years now in an Airbus and now 757. It doesn't use batteries, it gets its ANR power from the normal plugs.
I don't know how much louder a 737 is than a 757 but I would think it would work fine.
Is it as quiet as a Bose + the u-fly mic? Nope. Absolutely not. But I found the whole u-fly setup horrible when in the right seat (I'm sure it would work better in the left)
I've also used the non ANR Telex 850. It pretty much sucks. There is a big difference between the perceived quietness between the two models. But even so, I nearly always ditch the headset above 18