Do you have some dremel or a friend with one? Since you have the board out now, there's no need to worry about damaging the PCB and the cart is deep enough so there shouldn't be much risk with damaging the front, unless you have really shaky hands or press down to hard on the dremel (let the tool do the work!).
You could try snapping the back of the cart entirely? If you have a padded clamp, put it over the middle screw area then just pull the edges of the back off. If you have a clamp but it isn't padded then you could just use a towel, or an old t- shirt to prevent damaging the cart.
Hell if your really at wits end, you could try a saw.
(There is actually some logic in that last statement.)
Cut out a block of wood that will fit the contact area, not tight, just enough to stay in there, then get another piece of would small enough to fit in the gap at the top of the cart, between the front and back (A steel ruler will do as well!), then cut from top to bottom, on both sides of the screw. Once that's done, you can easily break the little bits left off. (See there was some *evil doctor-esque* logic there!).
A dremel is really your best bet now. Just cut around the screw and your home free! You can use a pair of pliers to crack the melted plastic around the screw off, then once your 3.8 secuirty bit has arrived, just unscrew the middle screw, put on new back (AFTER YOU CLEAN THE CONTACTS AND CART!), screw in the old screw and 2 new replacement screws and then your done!
