?? Trying to mentally picture what the difference is between my idea and what you have been doing as to how the lid opens.??
Ahh, "Do your lids lift off instead of hinges?".
Ok, "How about a Hybrid of my idea?
With bags sideways as I previously suggested, attach what are (now) the lower d-rings to inside handles of the lid and connect each bags (now) upper d-rings to each other via strap across/under the seat.
Does this seem like it might work?
Freelancer