From the bare metal leading edge of the horizontal stabilizer I would assume, it has a bleed air anti-ice system. Can somebody confirm the ice protection system design? Is the AoA Vane/pitot/static heating monitored/indicated? Is the elevator hydraulically actuated? The pure elevator related flight path fluctuations in combination with a straight track indicates to me that the issue was somehow related to pitch control, either due to the auto pilot, the FWB or mechanical/aerodynamical horizontal stabilizer/elevator issues without proper feedback to the systems/pilots.
I would assume the autopilot would quit if the pilot fights against it or some system malfunctions.