How large do you need the fabric like pieces to be?
Wet felting probably isn't going to be the solution, but I've made thin needle felted fabric. Here's what you do:
Lay out a THIN layer of fiber larger than you need the finish fabric to be.
On top of that lay out an even thin layer of fiber that has been cut and carded (1" - 1.5" in length).
Tack the two layers together - Tack (don't thrust your needles deep into these two layers.
Gently lift all the fiber off your work surface (lift in the direction of that first layer of long fibers) and turn it over.
Tack all the fibers from the other side.
Start surface felting at shallow angles to the fiber, turn it over as needed.
Hold the fabric you are creating up to a light to see which area might be to thin - add more fiber in those areas - tack and then surface felt.
It's possible to make very thin needle felted fabric this way. I work only on Clover brush mat - I do not know if this process might be a bit more difficult on foam work surfaces because of the "bounce" that they have - should still work.