All the waist belt does is to hold the pack tight against you so that some, or all, of the weight is borne on the circumference of your hips. Parachute cord would do more-or-less the same thing as a 3" wide web belt just not nearly as comfortably.
A pack sags because it is not stiff enough between the hip belt and the shoulder strap attachment. That is why heavier packs have frame sheets or stays of a wide variety. Stiffening a frameless pack can be done, within limits, by using your sleeping pad as a frame sheet or by installing a hoop stay such as is used with the old Six Moons Starlite pack.
I installed a SMD hoop stay in a G-4 that I made. It helped. But, frankly, the G-4 is not one of the most comfortable packs I have made.
In my experience, a frameless pack needs to be closely matched to the volume of its contents to cary well. The contents are, or should be, part of the pack suspension system. Otherwise, pack design is not particularly relevant to carrying comfort and a burlap bag would serve the purpose almost equally well.
The biggest problem with the G-4 pack and others of its ilk is that it has a large volume so a lightweight load fits inside rather loosely. This looseness allows the pack to fold in the middle somewhat when the hip belt is fastened and allows the pack to slump transferring some weight to your shoulders. To stiffen the pack, temporarily, try cutting two pieces of wood lath to the same length as your sleeping pad is wide. Then, fold them into your pad before you put it into the pad pockets. These wood strips will serve as a temporary (or permenant) stay and may increase the carrying comfort. Keep us advised.
May I walk in beauty.