You know... not all neoprene is closed-cell foam. You can get it in solid sheets too. About the foam: I guess the thinking in an application like this is that solid rubber is too hard. Is that right? Nobody's said it, but why else use a foam? In the case of wetsuits it's because of the insulating properties. I don't believe that's what you're trying for here.
If the foam is too soft, of course it will totally compress and it will be as if no foam was used; just solid rubber.
For reference, and since I could, I checked the tires on my Lotus for their hardness, and on the Shore A hardness scale, (typically used for elastomeric materials like rubber) I measured ~62.
On my wetsuit that I use for sailing, I measured 10. This is a linear scale. The measurements aren't exactly according to engineering standards (e.g. the neoprene foam is not .5" thick, and is nylon-covered), but the comparison is somewhat meaningful.
Hopefully this will confuse you about what to do. :-)