4 slings & ridge lines
What about using 4 slings & ridge lines?
Two slings above & two below where you want the spreader bar. Each end of the spreader bar connects to a high & a low sling at roughly 45 degree angle vertically (90 deg if you look from the side) and 60 degrees horizontally (if you looked top down). If you have ridge lines on both hammocks you shouldn't see (much) sag when one occupant gets out since the empty hammock's ridge line would go under tension and keep the full one from sagging and the low sling on the empty side would keep the full hammock from dropping toward the middle.
And you avoid needing poles along the tree.