Hi Deb,
What I would try, is with that shape youve got, use an Inset on the whole Mesh to get holding Edge Loops all around the top and bottom. Then you can remove your Bevel Modifier and let the Subdiv Modifier round out those sharp edges. The order should be Mirror, Solidify, Subdiv (and the Smooth By Angle).
So you get something like this (make sure that you have Clipping enabled in the Mirror Modifier, before Insetting, or Apply the Mirror, and re-Add one again after the Inset):

Don’t look at the Topology in this one, this is just a demonstration of the Edge Loops.