Water bath issue when doubling a cheesecake recipe to make bars
The water bath for a cheesecake is to control the temperature of the thick custard in the springform pan (cheesecake is technically a custard) - you don't need to worry about the moisture of the oven in the absence of a water bath.

The equivalency you stated of your pan volumes is a problem, though.    The two pans are not as comparable as your volumes suggest.  The area of a 9" round pan (your springform) would be π x r² = 3.14 x (4.5")² = 64 in.² (roughly).  The area of a 9" x 13" rectangular pan would be 9" x 13" = 117 in.² which is roughly twice the area of the springform pan.  What that means is that the same volume of cheesecake filling would fill a 9" x 13" pan roughly half as full as a 9" springform pan.

Unless you have a huge amount of filling, the custard should set-up much more quickly in the larger pan and should not be as much of a problem to bring-up to temperature evenly like a thicker cheesecake in a springform pan.  As long as you aren't planning to bake your bars in a scorching-hot oven and you remove them from the oven before they look fully set, you won't need the water bath.