I have shipped 3 glass bezels and never had a problem. I did use FedEx and marked the package FRAGILE - GLASS. Other then that I have a nice sturdy cardboard shipper folded in half. When I place the glass in there I make sure it has 4 folded pieces of sturdy cardboard folded in half and cover each edge of the glass with it. Sort of makes it "float" in my cardboard shipper.
I just throw it in a box about the size of the bezel(close is cool) with no packing and hope for the best. For people that i like, i pack it in a 3-4" box with rigid styrofoam, i talk to and caress it gently, then send it off with the utmost tlc.
You could always take it to a UPS store and have them pack it like a picture. The store I used to go to had thin boxes that would be about the right size. Just pack the edges with styrofoam (the sheet kind not just peanuts) and wrap the top and bottom with bubble wrap & then a layer of cardboard and then more bubble wrap.
I've had two come this way with absolutely no issues. I've also had 3 come just thrown into a knocked down box with little or no bubble wrap and no stytofoam. 2 survived intact and 1 is a box of shards.