The mass of the bipod will cause the front of the stock to slap up and down. It wore a nice rub mark on the underside of my 10FP barrel, so I finally changed stocks. But causing the stock to shift on the gun further back? That makes no sense to me unless the stock is flexing and deforming all the way back there just from the pull on the sling swivel. Without seeing and feeling the thing personally, I can't make a very intelligent guess about it.
One other thing you could try is to take a 10-15lb weight and hang it off the front sling swivel. I set a bag under the magazine well of the gun and then hold the toe of the butt down against the work surface with your hand to make the sling swivel suspend the weight. See if that puts the wiggle back into it. If so, the deformation theory seems verified. The heavier weight should exaggerate it, if that's what's happening.
The other reason I suggest so much weight is that you can pre-cut some shim strips of cardboard to slip between the barrel and stock near the front while the weight is pulling down. Then when you release the weight, the stock springs up against the barrel via the shims, applying pressure to the barrel at that location. This forces the 3rd harmonic vibration node to that particular location. It is what Harold Vaughn calls "O'Connor Bedding" because he thinks he first heard of it in a Jack O"Connor article that endorsed the method. Vaughn says it's the only form of bedding he's ever seen consistently improve non-benchrest rifles.
With wood stocks, the shims are ideally 120° apart, and are moved for and aft to tune the load. Once the sweet spot is found, release agent is put on the barrel (Pam spray works fine) at the shim location, and the shims are saturated with about any brand of slow set epoxy. I like both System Three T-88 and West System G/flex epoxy for this kind of work. The latter may have an edge with a plastic surface because of its slight flexibility. For me, its shelf life has been a little better.
Below is an illustration of the principle. In practice the shims can be any base material that can be glued. Balsa wood is fine. It can also extend all the way under the barrel and up the other side if that proves more convenient to do. I recommend finding an OCW sweet spot by Dan Newberry's method, then tuning the O'Connor bedding to that load, in particular.