Can't remember the tech side of the B737 very well but could it be pilot induced oscillation brought about by over correcting and breaking out the spoilers on the down going wing?
Agree. Have seen this (PIO) countless times in the simulator and not necessarily only when gusts fed in. One way the instructor can demonstrate this is by setting a visibility of 4 kms for an ILS. As soon as the approach lights appear in the gloom and no visible signs of a horizon I estimate 90% of pilots start the wing waggling which only get worse. The fix is to momentarily relax the death like grip on the control wheel and the problem goes away.
Had one very experienced pilot caught in the simulator with PIO and eventually he was pedalling the rudders as well and he crashed in the go-around. He accused the instructor of putting in un-briefed windshear. But the wind was zero. Tried it again and sure enough as he saw the approach lights he was off into severe PIO. He was quite convinced the instructor was pulling his pisser and sneaking in wind-shear.
Now this bloke had a real problem with temperament and he began to lose his cool. So the instructor put him on a ILS five mile final CAVOK and lo and behold no signs of POH. When the instructor then told him the wind was unchanged - that is no wind at all - the pilot refused to believe it was his own fault. You get those personality types occasionally.