Since metahuman facial animation is mostly based on bone positions, you likely will need to change bone transforms to match your morph target in some way. Deformer graph doesn’t allow you to change bone transforms, it can only read bone transforms and other bone related custom animation attributes (where you can potentially compute some type of offset using a custom control rig node) and deform vertices using your own custom kernel as none of the built-in deformer graph would do what you want here. However, computing such offsets and passing them through to the deformer graph as animation attribute is essentially recreating & retargeting the facial skeleton, which is can turn into something complex easily, so it might not be the right tool for the job. You might notice that it is not just the eyes that have the deformation issues, you might also need to adjust for lip area as well.
UE’s Mutable system on the other hand does offer some in-game bone-adjusting functionalities that may or may not help. Mutable can adjust selected bone positions following mesh morphs (Mesh Morph node) or following a procedural morph (Reshape Morph node). But given metahuman’s riglogic (bone deformation system) being a blackbox, the current bone movement formula might not exactly fit your needs, so you might have to tweak it to your specs. Is it something that you are already using? Did you find any limitation with it that is blocking you from achieving what you want that motivates to use deformer graph?