To reduce or prevent brass bouncing back in to the action from a hard case catcher, see U.S. Patent No. 6,836,991 (which is now expired). I modified an E&L hard plastic catcher for an HK 91 similar to that design and just put the wedged foam piece on the wall directly across from the ejection port, not on all the walls. Works well for me.
For the Uzi I use a M16/AR-15 mesh bag mounted "backwards" - with the Velcro strap behind the rear sight around the (folding steel) stock. I have a bunch of small rare earth magnets slid around the inside of the fabric on the bag wire to stabilize the bag against the of the receiver. Also works well for me.
YMMV, etc.
Best of luck to all, whatever you decide.