I tried ccd and many configrations of collisions and physics options. The shaking happens when char is strong enough to move the physics object a little bit, maybe centimeters.
that shake when the char starts pushing is usually the capsule fighting the cube, not the cube settings. capsule is kinematic so it wins every depenetration step and the cube pushes back next physics tick, you get that cm-scale buzz.
since it stops with animbp off, check the mesh first. mesh should be NoCollision (query only if you need traces), only the capsule blocks the physics object. if the skeletal mesh has collision enabled the anim pose moves it into the cube every frame and the solver never settles. also check the animbp isnt driving root motion into the capsule while movement mode is walking.
then tune the interaction itself: CharacterMovement has Enable Physics Interaction plus Initial Push Force Factor and Push Force Factor. lower them a lot for a test (like 10-20) and see if the shake drops. on the cube side turn on CCD, add a bit of linear damping (0.1-0.3) and angular damping, and keep mass sane vs the 88kg default char (a 10kg cube jitters more than a 50kg one). also bump Position Solver Iteration Count one step in project physics settings if you lowered it.
this pawn playlist covers the game thread vs physics thread split and why push force has to live on the right side, second half is exactly your case: Networked Physics - Pawn Tutorial
and the gameplay one builds pushing and pulling on top of it: Networked Physics - Gameplay Tutorial
