Notice that the spreader bar allows the middle of the hammock to sag further, and sets the edges to be higher. That is what happens with a line through the channel, but in an uncontrolled way, depending as it does on the occupation of the channel and the compression of the fabric
Since you already have the apparatus set up, and the channel line looks very strong, I suggest the following experiments, varying the amount of line and sag between the spreader bar and the channel.
Just carefully insert toggles, in the form of carabiners, midway through the cord on the suspension end at marked and varying lengths. You could use knots, and I'm not worried about strenghth, but the toggles will be easier to remove.
Your weight in the hammock will pull the line through and the carabiners against the spreader bar.