Timberr, I'm assuming you want to use the spreader bars with your Chrysallis bridge hammocks. The spreader bars will function best at the end of each hammock, rather than at the tree. The forces on the spreader bar will be less at the hammock ends. If you try to put them at the tree, as in your diagram, the forces will be much greater, hence the poplar dowels failing.
The downside to having the bars at the hammock ends is that one occupant transfers his/her motion to the other hammock. Once both occupants are settled in, it's no big deal in my experience.
To place the spreader bar at the tree, the bar will need to be able to withstand lateral stress more so than compression. Tubing tends to withstand these forces better than solid material The thicker the wall thickness of the tubing, the stiffer the tubing will be.
Carbon fiber withstands compression quite well, but I'm not sure it would hold up to the lateral forces involved if placed at the tree.