I may be visualizing the display wrong...I'm visualizing foam cut-outs under a piece of velvet covered cardboard, that would leave the dice in contact with the foam. If the cutout is totally velvet lined, I'd probably try it.

As for the dice storage tubes, do you have a local pharmacy? They'd probably sell you a few vials and lids at a pretty reasonable price. If they happen to be amber colored, well, that would help filter out some light.