Shooting off a bench and a bipod (never prone), I too find light wood or fiberglass-stocked rifles need either a lighter touch or for me to sorta grab the top of the scope to stop it hopping. With a Harris bipod, at least.
I might just suck, however.
My heavy KRG W3 chassis is way easier to shoot from the bench + bipod, and my AIAX is even easier for some reason.
The W3 and light rifles are usually easier to shoot well in an X-shaped heavy bag, sorta like a sandbag but mine isn’t full of sand.
The AI seems to shoot well no matter the rest. It’s a little eerie.
One thing I’ve noticed is I seem to usually shoot worse with a rubber-footed Harris with a light rifle. Bounces around more. Only use a Harris because my lighter rifles have one sling stud up front and using the Harris is convenient.
With those light guns, if I use a sling stud to picatinny adapter and then attach an Atlas SCAL with ski feet, I do a more of a free recoil technique (still use a rear bag). My results seem better this way, but occasionally the stars align and I shoot well with the Harris.
I’m thinking that since I do more dot drills than groups, the slight shifting I have to do when aiming for each new dot induces some springy torque in the Harris. Which then messes my shot up if I’m not perfectly aligned on the bench.
ANYWAY, with heavy rifles I just load the bipod a bit and get great results. The bipod you use has to be up to the task; I use an LRA Light Tactical Fclass (my favorite) and an Atlas SCAL, mainly. Sometimes with ski feet on the SCAL and lately with spikes on either so I can load them to learn something differen.
I do have an Elite Iron panner bipod and while it’s superior to all in getting a natural point of aim, it fits only one or two of my guns and is a pain to get all set up. The panner bit makes leaving the bipod on a no-go as it spins about and will scratch the barrel all up.
Hope some of this helps.