Keith Newton
I would lean more toward using a caulk, instead of fasteners. I would start by cutting the tip, at an angle with a fine little hole. You should be able to get a little between the loose parts, which would make a flexible little shim, if you allow it to dry before moving it around much.

The panel is normally left loose, so it can expand and contract with changing humidity, so you don't want to do anything to trap the side edges, or it will cause the panel to split when it tries to contract. So whatever you do, start out doing the minimum, like fastening the middle of the top and bottom.

Other options, is to make sure there are some of the little door bump pads on the back of the swing side, so it isn't just wood slamming against wood.

If these are concealed Euro hinges, they can be replaced with soft-close hinges, and for some, this can be an add-on which just snaps on top of the existing hinge.