I believe you will have less of a flat lay or a very tippy hammock. Rope, chain, hammocks will have a catenary curve when hung by their ends. To even get somewhat flat, you would need to pull the hammock very tight and you still won't get it flat. Add your weight to the system and even less so. The spreader bars will not affect this curve much. They are there to spread the material, less shoulder squeeze. It would have the same lay as your gathered end but not being able to lay on the diagonal.
The bridge hammock also has to deal with these laws of physics. The suspension, not the hammock body gets the curve. The hammock body is then hourglass cut to match the curve causing a flat lay. Think of suspension bridges (hence the name bridge hammock). The support cables are curved with the road bed hanging flat from that curve.
To get a somewhat flat lay the spreader bars in a non-bridge style hammock need to be pulled very tight. When laying in the hammock, the center of gravity will be higher relative to the support anchors, causing the "tippiness" of such a hang. In a bridge hammock or a gathered end hammock hung with a good amount of sag, the center of mass is well below the anchors, making them stable.
To get a really flat lay, you will want to lay parallel to the suspension. A gathered end hammock is really a bridge hammock turned 90 degrees, more of a cable stayed bridge design. Laying diagonally in a gathered end gets you close, a bridge hammock is laying parallel to the suspension. Of course, the hammock body needs to be cut to match the curve that the hammock wants to become to get a really flat lay.