you have to understand the difference between the displays aspect ratio and the source's aspect ratio. the jvc will output 16:9 all the time. that's what you want it to do. just like a flat screen, it's screen doesn't change shape when playing a BD to a different aspect ratio, it simply displays black bars top and bottom.
the same will happen with the jvc. when watching your 2.35:1 source, you'll get an image projected that is the same width, but not as tall. if the projector wasn't so good as showing black, you'd see that it's still projecting a 16:9 image, just with black bars top and bottom(like a flat screen).
there's two things you can do:
1. you can go with a 16:9 format screen, and never have to adjust anything. the biggest image will be when you watch 16:9 sources, and when you watch 2.35:1 sources the image will become shorter, leaving black bars on the top and bottom.
2. you can go with a 2.35:1 format screen, and use the lens memory feature to achieve 'constant image height'. what this means is that when you watch a 16:9 format source, the projector's 'full' image will fit on the screen with no projected image on the left and right edges of the scope screen. kind of like 'black bars' except that it's not the projector displaying black, it's no light being displayed there at all. then for 2.35:1 format sources you zoom in the projector so that it's projected 16:9 image spills over the top and bottom of your screen and fills it up completely horizontally. the part that spills over the top and bottom will be the 'black bars' and you won't see them at all.
i'll see if I can do this with some pictures. unfortunately i'm going the other way, using a 16:9 screen, but I think you should get a better understanding of what zooming in with the projector(that is still projecting 16:9) can show what looks like a full size 2.35:1 aspect ratio image.
here is my 16:9 screen
here it is with 16:9 content, nothing confusing yet...
now here is the lens shift guide on the jvc. the grid lines show a 2.35:1 AR in the middle, 16:9 around the outside, and where 'overscan' would crop the image if you used it(you shouldn't). as you can see, I have the 16:9 set here, the outermost lines hit the edge of my screen.
now, when i'm watching 2.35:1 content, I would adjust so that the 2.35:1 guidelines hit the edge of my screen(I roll my 16:9 up part way). you can even see the top edge of the projector's image shining on my ceiling
and here is the 2.35:1 content showing
and finally with my screen rolled up to match. notice you can't see anything above or below the screen, but the projector is in fact displaying 'black bars' above and below.
my recommendation is to get yourself familiar with your projector, and go with a 2.35:1 screen and use lens memory to achieve constant image height. you sound similar to me, I find 16:9 images too tall when they are the same width as my preferred size 2.35:1 image.