There are some racks that mount directly onto the seat tube and utilise a quick release clamp; they only take a small load tho.
This would avoid clamping onto the carbon. As others have hinted its not such a hot idea to do this as carbon is not designed for that kind of compression. You can find evidence of this with car racks where carbon frames are not recommended for down tube clamp racks.
You should be able to get some p clips that go onto the skewer but may require modification for your drop outs.
That chain look like its fit for the bin.