I'd sure try the silicon before the glue.

If the doors are plywood and MDF, glue is fine. Those two materials do not have a lot of movement.

If the stiles and rails of the doors are solid wood and the panels plywood, same thing applies.

But, the silicon is not nearly as permanent, so trying that is preferrable to me.

Try to make sure the panels are in the middle of any side to side play.

At that point you can fasten them in the middle top and bottom using just about anything you want to try, even brads (small nails) on the back of the doors.

A drop of glue top and bottom would also work.

I would drive very tiny shims (like toothpicks tapered to nothing with a razor knife) at three points on the back side, across the top and bottom of the panels. You could put a bit of glue on one side of these tiny shims. (makes more sense to glue to the panel than the frame). If they still rattle, add another to the middle of the panel edge along the vertical length.
Casey

Thanks for the ideas. I'll post an update when I figure out what works best.